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?

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pedrocarratore pedrocarratore · Answer 1 · 2019-11-30T09:35:25+01:00

Answer:

-First player wins:

[tex]P(HT) = p-p^2[/tex]

-Neither player wins:

[tex]P(HH) + P(TT)= 2p^2 -2p +1[/tex]

Step-by-step 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':

[tex]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\\[/tex]

Step 4: Add probability of each event

-First player wins:

[tex]P(HT) = p-p^2[/tex]

-Neither player wins:

[tex]P(HH) + P(TT)= p^2 + 1 - 2p +p^2\\\\P(HH) + P(TT)= 2p^2 -2p +1[/tex]