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Chapter 9: Problem 104

Probability In an experiment, three people toss a fair coin one at a time until one of them tosses a head. Determine, for each person, the probability that he or she tosses the first head. Verify that the sum of the three probabilities is 1 .

Short Answer

Expert verified
The normalized probabilities that the first, second, and third person tosses the first head are approximately \(0.5714\), \(0.2857\), and \(0.1429\) respectively.

Step by step solution

01

Determine Probability for the First Person

The first person will toss the coin first. The probability of tossing a head (success) in a fair coin is \(0.5\). Therefore, the probability that the first person tosses the first head is \(0.5\). This is derived from the formula of geometric distribution when the number of trials (n) is 1.
02

Determine Probability for the Second Person

The second person tosses the coin only if the first person did not toss a head. Thus, this is the second trial but the first for the second person. The probability of the first person not getting a head is also \(0.5\). Therefore, the probability the second person tosses the first head is the product of the probability that the first person does not get a head and the probability that the second person gets a head, i.e. \(0.5 * 0.5 = 0.25\).
03

Determine Probability for the Third Person

The third person gets a turn to toss the coin only if both previous people did not get a head. The probability for each is \(0.5\), and these are independent events. Therefore, the probability that the third person tosses the first head is the product of probabilities that both previous people don't get a head and of the third person getting a head, i.e. \(0.5 * 0.5 * 0.5 = 0.125\).
04

Verify the Sum of the Probabilities

Lastly, check if the sum of the probabilities calculated for each person equals 1. Add all these probabilities: \(0.5 + 0.25 + 0.125 = 0.875\). However, this sum is not equal to 1. The reason being, there is a remaining \(0.125\) probability that none of the three people get a head in their first round, and the cycle continues. In that case, the sequence of tossing starts again in the same order. Hence, the cycle continues infinitely until a head appears and the series is geometric. Therefore, we divide each probability mentioned above by \(1-0.125=0.875\) to normalize them. This normalisation guarantees that the probabilities add to 1.
05

Normalizing the Probabilities

After normalizing the probabilities, the outcome for each person is: First person: \(0.5/0.875 = 0.5714\), Second person: \(0.25/0.875 = 0.2857\), Third person: \(0.125/0.875 = 0.1429\). Sum these values. \(0.5714 + 0.2857 + 0.1429 = 1\). As expected, sum of the probabilities is equal to 1.

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