Probability of a Coin Toss

Question

A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?

Answer 1

The possible failure outcomes are as follows:

TTH, THT, HTT, HTH, and TTT. The only other three possibilities are HHT, HHH, and THH; these are all success events, since each has at least 2 consecutive heads.

The theoretical probability of each of these events is (1/2)³ = 1/8. Since these are all independent events, the probability of one of these events occurring on three consecutive flips is the sum of these, or 5/8.

So, the probability of at least 2 consecutive heads is 1 - the probability of that not happening, or

1 - 5/8 = 8/8 - 5/8 = 3/8.

Answer 2

SOLUTION 1: There are  ways to flip the coins, in order. There are two ways to get exactly two consecutive heads: HHT and THH. There is only one way to get three consecutive heads: HHH. Therefore, the probability of flipping at least two consecutive heads is .

SOLUTION 2: Let's use complementary counting. To start with, the unfavorable outcomes (in this case, not getting 2 consecutive heads) are: TTT, HTH, and THT. The probability of these three outcomes is , , and , respectively. So the rest is exactly the probability of flipping at least two consecutive heads: . It is the answer .