Finding the Expected Number of Flips in Coin Tossing Problems

 A coin, having probability p of landing heads, is continually flipped until at least

one head and one tail have been flipped.

(a) Find the expected number of flips needed.

(b) Find the expected number of flips that land on heads.

(c) Find the expected number of flips that land on tails.

(d) Repeat part (a) in the case where flipping is continued until a total of at least

two heads and one tail have been flipped.

Coin tossing is a classic example of a probability problem and is often used in introductory statistics courses to illustrate key concepts such as expected value. In this blog post, we will examine a coin tossing problem where the goal is to find the expected number of flips needed to get at least one head and one tail.

(a) Find the expected number of flips needed:

Let's denote the expected number of flips needed as E(N). The first flip can either be heads or tails. If the first flip is heads, the second flip must be tails, and the expected number of flips will be 2. If the first flip is tails, the second flip must be heads, and the expected number of flips will also be 2. Hence, we have:

E(N) = (1 - p) * 2 + p * 2 = 2 * (1 - p + p) = 2 * 1 = 2

(b) Find the expected number of flips that land on heads:

Let's denote the expected number of flips that land on heads as E(H). If the first flip is heads, the expected number of flips that land on heads is 1. If the first flip is tails, the expected number of flips that land on heads is 2. Hence, we have:

E(H) = (1 - p) * 2 + p * 1 = 2 - p + p = 2

(c) Find the expected number of flips that land on tails:

Let's denote the expected number of flips that land on tails as E(T). If the first flip is heads, the expected number of flips that land on tails is 1. If the first flip is tails, the expected number of flips that land on tails is 2. Hence, we have:

E(T) = (1 - p) * 1 + p * 2 = 1 - p + 2p = 1 + p

(d) Repeat part (a) in the case where flipping is continued until a total of at least two heads and one tail have been flipped:

Let's denote the expected number of flips needed as E(N2). The first two flips can either be heads or tails, and the third flip must be tails. If the first two flips are heads, the expected number of flips is 2. If the first two flips are tails, the expected number of flips is 3. Hence, we have:

E(N2) = (1 - p^2) * 3 + p^2 * 2 = 3 - 2p^2 + 2p^2 = 3

In conclusion, finding the expected number of flips in coin tossing problems is a straightforward application of expected value. By using the formula for expected value, we were able to find the expected number of flips needed, the expected number of flips that land on heads, and the expected number of flips that land on tails. We also saw how to modify the problem to find the expected number of flips in a slightly different scenario.