Question

Problem 17.4. To determine which of two people gets a prize, a coin is flipped twice. If the flips are a Head and then a Tail, the first player wins. If the flips are a Tail and then a Head, the second player wins. However, if both coins land the same way, the flips don’t count and the whole process starts over. Assume that on each flip, a Head comes up with probability p, regardless of what happened on other flips. Use the four step method to find a simple formula for the probability that the first player wins. What is the probability that neither player wins?

Answer 1

-First player wins:

`P(HT) = p-p^2`

-Neither player wins:

`P(HH) + P(TT)= 2p^2 -2p +1`

Explanation:

Applying the 4 step method:

Step 1: Identify events for which probability is to be determined:

- First player wins: First coin is heads, second coin is tails (HT)

- Neither player wins: Both coins are heads or both coins are tails (HH or TT)

Step 2: Calculate total number of possible outcomes:

Four possible outcomes: HH, HT, TT, TH

Step 3: Calculate probability of each event

Since the probability of a coin coming up heads is 'p':

`P(HH) = p^2\\P(TT) = (1-p)^2 = 1 - 2p +p^2\\P(HT) =p*(1-p) = p-p^2\\P(TH) = (1-p)*p = p-p^2\\`

Step 4: Add probability of each event

-First player wins:

`P(HT) = p-p^2`

-Neither player wins:

`P(HH) + P(TT)= p^2 + 1 - 2p +p^2\\\\P(HH) + P(TT)= 2p^2 -2p +1`