Probability: Putting Numbers on Uncertainty

1.What Probability Means

When all outcomes are equally likely, the probability of an event is "the number of outcomes in which the event happens ÷ the number of all possible outcomes." The probability of rolling an even number with one die is the 3 even outcomes (2, 4, 6) divided by all 6 outcomes: 3/6 = 1/2.

Probability is always between 0 and 1. 0 means it never happens, and 1 means it is certain to happen. A probability of 1/2 is not a promise that "it will happen once in every two tries"; it means that if you repeat it very many times, the share of times it happens gets close to 1/2. Getting 7 heads in 10 coin tosses is not unusual, but with 10,000 tosses the share of heads usually clusters around 0.5.

2.Counting Outcomes: The Multiplication and Addition Rules

How many outfits can you make from 3 shirts and 4 pairs of pants? For each shirt you can choose from 4 pairs of pants, so 3 × 4 = 12 outfits. When choices are linked by "and" like this, you multiply the numbers of outcomes (the multiplication rule). A 4-digit numeric passcode has 10 options for each digit, so there are 10 × 10 × 10 × 10 = 10,000 possibilities.

On the other hand, when outcomes are linked by "or" and the two events cannot happen at the same time, you add the numbers of outcomes (the addition rule). On a die, "2 or less, or 5 or more" is 2 outcomes + 2 outcomes = 4 outcomes. But if the two events can overlap, you have to subtract the overlap once. "Even or a multiple of 3" is 3 even outcomes + 2 multiples of 3, minus the 6 that is both, counted once: 4 outcomes (2, 3, 4, 6).

You also need to consider whether order matters. There are 5 × 4 = 20 ways to choose a president and a vice president from 5 people, but to choose 2 representatives without order, (A, B) and (B, A) must count as the same, so there are 20 ÷ 2 = 10 ways.

3.Complements: Count "At Least Once" Backward

Counting "the probability that it happens at least once" directly is complicated, because you have to consider every case: happening once, twice, three times, and so on. In that situation, it's simpler to find the opposite, "the probability that it never happens," and subtract it from 1. The event that a given event does not happen is called its complement.

4.Independent Events: Multiply the Probabilities

Two events are independent when the outcome of one does not affect the probability of the other. A coin toss and a die roll don't affect each other, so the probability of getting heads and also rolling a 6 is 1/2 × 1/6 = 1/12.

When events are not independent, the conditions change before you multiply. If you draw balls from a bag with 3 red balls and 2 blue balls without putting them back, then once the first ball is red, the probability that the second is also red is not 3/5 but 2/4. So the probability that both balls are red is 3/5 × 2/4 = 6/20 = 3/10. Real-world events often look independent but are actually tangled up with each other, like the weather in the same area on the same day.

5.The Gambler's Fallacy

You toss a coin and get heads 5 times in a row. Is tails due next? No. A coin has no memory. If it is a fair coin, the probability of heads on the sixth toss is still 1/2. Believing that "the other side is due now" is called the gambler's fallacy.

It's confusing because two questions get mixed up. "Before tossing, the probability of getting 6 heads in a row" is small: (1/2)⁶ = 1/64. But "given that 5 heads have already come up, the probability that the next toss is heads" is 1/2. What has already happened is no longer a matter of probability.

Lottery numbers, "numbers that are due" in a lotto draw, and the idea that "it's my turn to win" after a losing streak are all the same fallacy. On the other hand, when outcomes do affect each other (as with the bag above, where balls are not put back), earlier outcomes change the next probability. The key is to first ask whether the events are independent.