1. What Is the Chance of Winning When 2 People Draw From a Lottery With 2 Winners in 10 Tickets?
Imagine a lottery with 10 tickets, where 2 are winners. You put the ticket back after drawing it. The chance of winning on one draw is 2/10, or 0.2. The chance that 2 people draw in a row and both lose is 0.8 multiplied by 0.8, which equals 0.64, or about 64%. This means that even with 2 people drawing, the chance that at least one wins is only about 36%. At this small number of tries, it is not strange if no one wins. So, on average, how many people does it take to get a winner?
2. Why an Average Win on the 5th Try Does Not Mean You Will Win Within 5 Tries
If you put the ticket back, the chance of winning stays the same every time. The average number of draws until the first win is 1 divided by 0.2, which is 5 [6]. For a die, the average number of rolls to get a 1 is 6. However, "average 5 tries" does not mean you are guaranteed to win within 5 tries. The chance of losing 5 times in a row is 0.8 multiplied by itself 5 times, which is about 0.33. So, about 1 in 3 times, you will not win even after 5 draws. If you do not put the ticket back, the number of losing tickets decreases, making it easier to win. There are 45 ways to arrange the 2 winners among 10 tickets, and the average position of the first winner is 11/3, or about 3.7 [7]. This is shorter than when you put the ticket back. This is an idealized calculation and does not guarantee actual lottery odds.
3. How Close to Half Were Buffon's 4,040 and Pearson's 24,000 Coin Tosses?
Buffon tossed a coin 4,040 times and got heads 2,048 times. The ratio was about 0.507 [4]. Pearson tossed it 24,000 times and got heads 12,012 times, a ratio of about 0.5005 [4]. These counts are reported in a JSTOR Daily article. Pearson’s ratio was closer to 0.5. However, the difference between the number of heads and half the total tosses was 28 for Buffon and 12 for Pearson. This difference went up and down. It is the ratio that gets closer, not the difference in counts. A study that tossed real coins 357,057 times reported that the chance of landing on the same side it started on is 0.508 [8]. In this study too, the bias was very small, and it was not a case where heads came up easily.
4. Is Tails More Likely After Heads Comes Up 4 Times in a Row?
With a fair coin, even if heads comes up 4 times in a row, the chance of heads on the 5th toss is still 1/2 [5]. The idea that "it is tails' turn next" is called the Gambler's Fallacy. Does this contradict the Law of Large Numbers? Sources say there is no principle that small numbers of observations must match the expected value, and that a streak favoring one side is not immediately canceled out by the other [3]. For example, if heads comes up 8 times in the first 10 tosses, the ratio is 0.8. If the next 990 tosses are exactly half heads and half tails, the total for 1,000 tosses is 503 heads, and the ratio gets closer to 0.503. This is a hypothetical calculation to see how the idea works, but the bias is not canceled out; it is just diluted.
5. Who Proved the Law of Large Numbers and Who Gave It Its Name?
The first strict proof of the Law of Large Numbers was by Jacob (Jacques) Bernoulli. He called it the "Golden Theorem" and is said to have worked on it for 20 years [2]. His book Ars Conjectandi was published 8 years after his death, in 1713, by his nephew Nicolaus [2]. The name "Law of Large Numbers" was given by Poisson in 1837 [3]. This law has conditions. Each result must not affect the others, and the trials must have the same probability [9].
6. Why Insurance Can Predict a Group's Pattern but Not One Person's Future
The system of insurance is said to be based on the Law of Large Numbers. We do not know when an event will happen to one person, but when many people are gathered, the ratios by age and gender get close to a constant value. This allows insurance premiums to be calculated [1]. In roulette, you cannot predict the win or loss of a single spin. However, it is said that the house's profit gets closer to a predictable ratio when the number of spins is high [3]. Even if individual results are unknown, as the group or number of trials increases, the ratios become visible.
7. How to Compare 10 Coin Tosses and 100 Coin Tosses in Your Own Experiment
Repeat this experiment 3 times: toss a coin 10 times and record the number of heads. Then toss it 100 times in a row and record the ratio of heads. Compare how the variation in ratios differs between the records of 10 tosses and 100 tosses. When tossing, use a height just enough to drop it to the floor, and do not throw it toward your face or eyes. If you want to check more, try tossing several sets of 10 and several sets of 100, and compare how the ratios of heads are scattered. Compare whether the sets with more tosses cluster around 0.5.
SAFETY NOTE: For safety, do not throw the coin toward your face or eyes.