DON'T BLAME THE MATH

a probability blog

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Tag: blog

  • THE TRUTH ABOUT THE SPOTIFY SHUFFLE

    A very common feature of Spotify is the Shuffle, this puts songs from a certain album/playlist in a different order than they are usually in. The entire purpose of the shuffle is to be random, but what if i told you the shuffle uses a system that is much less random and more systematic, but makes it feel much more random.

    Lets take a simple example that shows the true meaning of randomness before we understand the shuffle system:
    If you asked anyone what the result would be if you randomly flipped a fair coin any given number of times, they would probably say an alternating sequence between heads and tails: HTHTHTHT……., and this would seem random. However, true randomness means you can constantly have consecutive heads or consecutive tails as every flip is independent to the one before it! Now lets take this same logic into the shuffle!

    Say I have a playlist with 100 songs, out of which 5 of them are made by Ed Sheeran. If the shuffle was truly random there is always a chance I will get all five of them in a row, but this wouldn’t feel random to an average listener, it would seem like a bug, and while this system is as random as it could get, it must be rigged to give variety and prevent such a situation from happening. Rather the system actually evenly splits certain genres or artists across the playlist. For example:
    Shuffle #1(One case for truly random, what listeners don’t like):
    Rock–>Pop–>Pop–>Pop–>Jazz
    Shuffle #2(Balanced and rigged, what listeners like):
    Pop–>Rock–>Pop–>Jazz–>Pop
    So if I have a case where 20% of my playlist is Country music, Spotify will map out the songs and space them exactly 5 spots apart, and will do this for all the other genres. The system I said would work, except for the fact that the shuffle on a certain playlist would be nearly identical everytime! The scientists who created this logic saw this through, so they introduced a tiny amount of random offset so the entire playlist isn’t completely in a pattern, so the country song could play in the 4th or 6th slot rather than exactly the 5th, ensuring a varied listening set everytime!

    So next time you listen to your favorite playlist using Shuffle, think back to the genius math behind it!

  • MONOPOLY probability(and how to use math to increase your odds at a win)

    The most famous board game of all time, MONOPOLY, is traditionally a game that has a history of going on for hours and a game that is entirely known to be played with strategy and a decent bit of luck, but what if I told you there was another factor? MATH!

    To start, lets breakdown 2 crucial details that will elegantly lead to a nice ranking of which color sets to aim for:
    1. Many would think that every MONOPOLY spot has an equal probability of landing on, but the reality is quite the opposite. The most landed on square is the notorious JAIL square. Let’s break this down:
    a.There is the self explanatory Just Visiting spot, this has the same probability as moving to any ordinary square.
    b.Then diagonally across the board we have the GO TO JAIL space, like before the same probability as any ordinary square
    c.Lastly there are also one chance and community chest card each titling GO DIRECTLY TO JAIL each with a 1/16 probability of drawing from their respective decks
    As you can see the JAIL square is superior(in terms of landing probability) to any other spot on the board, in fact this entire post centers around this very concept!

    2.The second detail all has to do with how the game is played, DICE! Well a normal dice rolls on any number evenly with each number a probability of 1/6 but like in MONOPOLY when there are two dice your expected value for each roll is a 7. In fact you have a 1/6 chance in doing so(this is calculated by 6 pairs making 7 and the total 36 number of outputs)!
    Now based on these two details let’s pair up frequency and price to get a overall ranking of the colorsets!

    EXPLANATION:
    (S)Orange: The colorset is clustered 6,8,and 9 spots away from the most visited spot jail, so from jail you have a 7/18 chance of hitting the orange set, it is also worth the same as the pink per house but has a much higher landing probability!
    (A)Light Blue: This colorset is severely undervalued! Its very easy to grab yourself hotels and is clustered 6,8,and 9 away from a common spot on the board GO!
    (A)Red:Expensive,but worth it. Around two moves away from jail as well as having cards like ADVANCE TO ILLINOIS AVENUE.
    (B)Yellow:Same price as red and in a nice spot on the board!
    (C)Pink:Two moves from GO, near jail as well!
    (D)Dark Blue: A super-pricey colorset with only two squares, high risk, high reward!
    (E)Brown: Completely irrelevant unless you stack houses/hotels, only two spots!
    (F)Green:This is statistically the worst colorset, houses are priced the same as Dark Blue, and there is also the GO TO JAIL square that can suck up traffic.

    Now this is all a general ranking depending on the game situation this ranking will obviously change, sometimes the Green colorset may win you the game, or the Orange will loose you the game, but in terms of math and probability these rankings are pretty accurate!

    So next time you’re playing MONOPOLY follow these general rankings to boost your odds at a victory, and leave the people you play with in shock!

  • What’s the Weather Like?

    You look at your forecast. 30% chance of rain, that’s a bummer(unless you’re one of the rare people who like the rain). But what does this number really mean?

    Many assume this number means that 30% of the given town would rain or it will rain for 30% of the day, but what it actually means is far more interesting and uses a clever calculation not many are aware of. When the forecast shows 30% it means that there is a 30% probability(I love that word) that at least 0.01 inches of rain will hit any given surface in the town over a certain time period(usually one,twelve,or twenty four hours). So now that we know what a 30% chance of rain means, how is it calculated?

    The National Weather Service says the probability is calculated like this, where P=probability of rain, C=confidence of prediction, and A=percent area of town expected to receive rain:
    P=C*A
    I know its pretty surprising how simple that is, you were probably expecting some crazy formula, or applying Bayes theorem and finding the expected value, but no its just multiplying two values!

    So for example, if P=30%(I don’t know why I keep using that number), you could have the following extremes:
    1. You have small storms incoming on only a fraction of the area but the meteorologist is very confident that this event will occur
    2. You may get a large storm flooding your entire town but the meteorologist is not super confident this will occur
    These both will show the same number on your weather app but they are two very different types of storms and occurences.

    So next time you look through the rain probability always think what kind of confidence or area of rainstorm you could be encountering!

  • THE PRISONER’S DILEMMA-a core situation of game theory and probability

    The Prisoner’s Dilemma is considered as one of the greatest situations in game theory, and like many problems it seems very simple yet the math and logic behind it is very interseting. The situation simply goes:
    Two partners in crime are arrested. The police put them in separate rooms and offer them the same deal:
    If you both betray each other, you both get 2 years.
    If you betray your partner and they stay silent, you go free and they get 3 years.
    If you both stay silent, you both get only 1 year on a minor charge.

    So what do you do? Now you would think that a huge part of this has to do with your relation to the other person and how strong your bond and loyalty is. While this does play a factor if you were entirely focused on a shorter sentence this social aspect is irrelevant.

    Let E(silent) be the expected value of prison sentence if you stay silent, and let E(betray) be the expected value of prison sentence if you betray your partner.
    Also let P(this is the variable, depending on the other person’s loyalty) be the probability that your partner stays silent, meaning 1-P is the probability that they betray you.
    Now let’s split into cases on if you stay silent or if you betray:

    Case 1: You stay silent–> if they stay silent as well you get 1 year, if they betray you get 3 years
    E(silent)=1*(P)+3*(1-P)=3-2P (if you are confused where this formula is coming from, it is simply just multiplying the outcome by the probability of it happening, this is the most straightforward way of calculating expected value)
    Sentence for you: 3-2P years

    Case 2: You betray–> if they stay silent you get 0 years, if they also betray you get 2 years
    E(betray)=0*(P)+2*(1-P)=2-2P
    Sentence for you: 2-2P years

    Conclusion: E(silent) is 3-2P years, while E(betray) is 2-2P years. As you can see you will always get 1 year less on your sentence if you betray!

    So the moral of the story is to never lie to police(and always betray your friend, just kidding), but like always don’t blame the math!

  • WORLD CUP!!!!! post 1

    Penalties. A “free” goal in a regular game, but in a tense shootout in the World Cup it could make or break your chances to win the trophy. So imagine this. You’re standing 12 yards from the goal, you need to make this or your team goes home, you have an entire country on your back, where do you aim?
    Now first instinct, the top corners, thinking its the hardest for the goalie to get to, but let’s break it down using statistics.
    This is a diagram made by me:

    red: minimal effort needed for save(red lines mark wing span of goalie)
    yellow: 50/50 requires good power to make, assuming goalie goes the right direction
    green: high probability of making(assume a make for calculation purposes)

    As you can see the corners, to no ones surprise are the best outcome, but which corner to aim at?
    I claim aiming for the bottom corners is better than aiming for the top, let’s break it down:
    When going for the top you have many things to worry about, for starters its common sense that shooting an aerial ball will be harder to aim than shooting a ground ball(because you have the “support” of the ground, and there is no force of gravity acting upon the ball). There is also the concern of hitting it too high or too low, when hitting to the top corners, let’s say this is your stats:
    50%: getting it in the box(score) expected score:1
    25%: hitting outside the goal(miss) expected score:0
    25%: hitting below or to the yellow side of the box(50/50) expected score: 1/2
    (these seem like bad stats, but remember it is much harder to shoot in the air rather than on the ground)
    Doing some math calculations(counting and probability is actually my favorite math subject) we see that the expected score is 5/8, meaning there is a 5/8 probability that you make it.

    This is not a great percentage in a high stakes game so let’s check out the other case, aiming for the bottom:
    Now for this case we don’t have to worry about hitting too low because unless your hitting into earth’s crust(which is an such an impressive feat that it definitely deserves a goal) there’s nothing lower you can hit, and since you are never taking the ball too high off the ground so we don’t need to worry about hitting too high:
    So let’s say these are your stats:
    85%:hitting in the box(goal) expected score:1
    10%:hitting outside of the goal(miss) expected score:0
    5%: hitting to the yellow side of the box(50/50) expected score 1/2
    Doing the calculations your expected score is 7/8 meaning there is a 7/8 chance of making.

    So let’s sum it up, a 5/8 chance of making(top corners), and a 7/8 chance of making(bottom corners), well the obvious pick is the bottom corners(these numbers will obviously vary slightly depending on the strength of the goalie and strength of the shooter)!
    Additionally, when picking which side to pick, you may want to go for the goalie’s non-dominant hand because he may have less power and/or control in that hand.

    Evidence, and a real-world example shows this. Harry Kane arguably the best current-day penalty shooter aims for the bottom corners the majority of the time!
    Summing it up, next time(if ever) you’re shooting a penalty, going for the bottom corner is the smartest choice.

    Thanks for reading, and enjoy the World Cup!

  • SUPER BOWL LX!

    The biggest game of the NFL season is here! Get your chips,guac, and pizza ready for a classic rematch from Super Bowl XLIX! But without further to do let’s get started with my predictions!

    Overall game trend: This game will be filled with thrillers and I’m predicted AT LEAST ONE TRICK PLAY.

    Team Analysis:
    SEAHAWKS:
    The Seahawks are the favorites to win this game on almost every sporting app. Needless to say, this team is GREAT. They have superstar offensive weapons such as JSN and Kenneth Walker, and Sam Darnold has been looking pretty good in the last two games! More than that, their defense is truly special. It has all round players that make it extremely difficult to score. All in all, it’s pretty hard to stop a team that’s good at almost everything.
    THE X FACTOR: RASHID SHAHEED, in both playoff games Shaheed has started cooking from the very start, including a punt return touchdown in the San Francisco game, and a 60 yard catch in the Rams game!

    PATS:
    Going in as the underdogs, and emerging common opinion is starting to form from many: The Pats have had a really easy road to the playoffs, and are actually much worse then they are said to be. I personally don’t believe this. While it is true that the Patriots have gone through an easy schedule, it is not to discount that this team beat out the Buffallo Bills and Josh Allen and won their division. Their playoff matchups might have been seemingly easy as well, but it is not to discount that they were able to rack up 5 interceptions in the game against the Texans. I think Drake Maye and the Pats offense can put up a serious contest against the hawks, an their defense has definietly been playing great as well.
    THE X FACTOR: MARCUS JONES, he is a very dynamic player, and a keystone to the Pats defense, if anyone can get an interception of Sam Darnold, I think it will be him.

    WINNER PREDICTION:
    Seahawks-31,Pats-27
    SEAHAWKS BECOME THE CHAMPIONS OF SUPER BOWL LX!
    I think the Seahawks are just too dominant for me to not pick them, I think the Pats will put up a fight, but at the end of the day there is only one winner, and I think the Seahawks take the Lombardi home this year.

    MVP Prediction:
    SAM DARNOLD
    I feel like it is hard to pick anyone but Quarterback for this award, but I truly think Sam Darnold is going to have a career-defining game this Sunday.

    Here is a banner to display my prediction:

    Image made by me

    P.S.: I am personally rooting for the Pats because I am a Niners Fan, and I don’t want the Seahawks to win a Super Bowl at Levi’s.

    Thanks for reading!

  • SUPER BOWL LX!

    The biggest game of the NFL season is here! Get your chips,guac, and pizza ready for a classic rematch from Super Bowl XLIX! But without further to do let’s get started with my predictions!

    Overall game trend: This game will be filled with thrillers and I’m predicted AT LEAST ONE TRICK PLAY.

    Team Analysis:
    SEAHAWKS:
    The Seahawks are the favorites to win this game on almost every sporting app. Needless to say, this team is GREAT. They have superstar offensive weapons such as JSN and Kenneth Walker, and Sam Darnold has been looking pretty good in the last two games! More than that, their defense is truly special. It has all round players that make it extremely difficult to score. All in all, it’s pretty hard to stop a team that’s good at almost everything.
    THE X FACTOR: RASHID SHAHEED, in both playoff games Shaheed has started cooking from the very start, including a punt return touchdown in the San Francisco game, and a 60 yard catch in the Rams game!

    PATS:
    Going in as the underdogs, and emerging common opinion is starting to form from many: The Pats have had a really easy road to the playoffs, and are actually much worse then they are said to be. I personally don’t believe this. While it is true that the Patriots have gone through an easy schedule, it is not to discount that this team beat out the Buffallo Bills and Josh Allen and won their division. Their playoff matchups might have been seemingly easy as well, but it is not to discount that they were able to rack up 5 interceptions in the game against the Texans. I think Drake Maye and the Pats offense can put up a serious contest against the hawks, an their defense has definietly been playing great as well.
    THE X FACTOR: MARCUS JONES, he is a very dynamic player, and a keystone to the Pats defense, if anyone can get an interception of Sam Darnold, I think it will be him.

    WINNER PREDICTION:
    Seahawks-31,Pats-27
    SEAHAWKS BECOME THE CHAMPIONS OF SUPER BOWL LX!
    I think the Seahawks are just too dominant for me to not pick them, I think the Pats will put up a fight, but at the end of the day there is only one winner, and I think the Seahawks take the Lombardi home this year.

    MVP Prediction:
    SAM DARNOLD
    I feel like it is hard to pick anyone but Quarterback for this award, but I truly think Sam Darnold is going to have a career-defining game this Sunday.

    Here is a banner to display my prediction:

    Image made by me

    P.S.: I am personally rooting for the Pats because I am a Niners Fan, and I don’t want the Seahawks to win a Super Bowl at Levi’s.

    Thanks for reading!

  • NFL HONORS!

    Here are my picks for each award this year:
    MVP: Matt Stafford
    Drake Maye makes a good case, and I am sure he will win one later in his career, but Matthew Stafford’s numbers this year are insane.
    DPOY: Myles Garrett
    There’s no question. If he doesn’t win it might be an even bigger scam than Bill Bellichik not being a first-ballot HOFer
    OPOY: JSN
    This one’s tight but JSN went off this season!
    CPOY: CMC
    He was one of the only starters on the Niners fully healthy this year, and acted as both a running back and a receiver, coming off an injury this is the best you can get.
    OROY: Tet McMillan
    Best receiver on the panthers by far, easily performed his pick number
    DROY: Carson Schwesinger
    Not much to choose from but, he did pretty good.
    Coach of the Year:Ben Johnson
    A lot of people deserve this, but the Chicago Bears turnaround and run was incredible this year.

    Thanks for reading!

  • NFL PLAYOFFS 2026! CHAMPIONSHIP SUNDAY!

    Here is my analysis both the conference championship games! Enjoy!
    AFC:
    Broncos vs. Pats
    Was a bit late to blogging, so the game is actually going on right now(it’s the first quarter). I think this game would be a blowout with Bo Nix under quarterback, however, I must admit, the Back up Stidham is not bad in fact I feel like he is playing very good considering he hasn’t thrown a touchdown since the literal first week of 2024! I think the Pats are just not looking as good and dominant as they were in the regular season. And for both those reasons, I see the Broncos actually coming out with an upset!
    MY PICK: Broncos-31 Pats-20, WINNER:BRONCOS

    NFC:
    Seahawks vs. Rams
    This is going to be a banger. The last time these two teams played it ended with a last minute 2-point conversion by Seattle to win the game with the score of 38-37. I see nothing less in this game, but think that the Seahawks defense will actually be able to hold of Matthew Stafford,Puka Nacua, and the rest of the Rams offense from breaking free. I for sure think it’s going to be a high scoring battle, but I think Seattle is going to win it.
    MY PICK: Seahawks-38 Rams-31, WINNER:SEAHAWKS

    This would be the Super Bowl LX match up with my predictions(image made by me)!

    Make sure to check out next week’s post which will contain my predictions on who will win the major NFL Awards(MVP,DPOY,OPOY,Coach of the Year,etc.)

    Thanks for reading!

  • NFL DIVISIONAL ROUND 2026!

    This season’s playoffs are off to a thrilling start last week there were 4 games that went down to the very last play! Even better, I predicted every single winner correctly, so let’s keep it rolling!

    Saturday:
    Broncos vs. Bills
    I think there is one guy who basically just does it for me. Yes, I am talking about Josh Allen, probably the best quarterback right now, sure he isn’t a favorite for the MVP award again, but he’s still so talented! I also think the Broncos and Bo Nix will be pretty rusty coming off of the bye, and feel like they will be underwhelmed with the playoff pressure.
    MY PICK: Bills-31 Broncos-17, WINNER: BILLS

    Seahawks vs. 49ers
    If you have been reading and following my posts you probably know I am a niners fan so I’m predicted them to win here, but I can 100% see Seahawks winning this. I just think that playoff Sam Darnold will come through and the Niners Defense will force a few turnovers
    MY PICK: 49ers-27 Seahawks-24, WINNER:49ERS

    Sunday:
    Pats vs. Texans
    If I had to make predictions on this game last week, I would have easily said the Pats take this home, but against Chargers, despite winning, they did look a little weak, and the Texans Defense is just too good.
    MY PICK:Texans-17 Pats-10, WINNER:TEXANS

    Bears vs. Rams
    What a way to finish of this week’s games. I know for a fact that this is going to be a scoring battle with Caleb Williams on one side, and Matt Stafford with his elite receivers (Puka Nacua and Davante Adams) on the other. I think the Rams will get affected by the weather, however leading to the Bears upsetting the Rams on yet another game-winning drive by Caleb Williams and Ben Johnson!
    MY PICK:Bears-38 Rams-34, WINNER BEARS

    THIS WOULD BE THE UPDATED PLAYOFF PICTURE WITH MY PICKS:

    I could see all of these games turning up differently, after all these are just predictions! But hope you enjoyed!

    Thanks for reading!