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Chapter 23: Problem 548

A coin is tossed 3 times, and 2 heads and 1 tail fall. What is the probability that the first toss was heads?

Short Answer

Expert verified
The probability that the first toss was heads, given that 2 heads and 1 tail fall, is \(\frac{2}{3}\).

Step by step solution

01

Identify the possible sequences

There are three coin tosses, with each toss having 2 possible outcomes (heads or tails). This means there are a total of \(2^3 = 8\) possible sequences. We will list all the sequences below: 1. HHH (Heads, Heads, Heads) 2. HHT (Heads, Heads, Tails) 3. HTH (Heads, Tails, Heads) 4. HTT (Heads, Tails, Tails) 5. THH (Tails, Heads, Heads) 6. THT (Tails, Heads, Tails) 7. TTH (Tails, Tails, Heads) 8. TTT (Tails, Tails, Tails)
02

Identify the sequences with 2 heads and 1 tail

Now we need to identify which of these sequences have exactly 2 heads and 1 tail: 1. HHT (Heads, Heads, Tails) 2. HTH (Heads, Tails, Heads) 3. THH (Tails, Heads, Heads) There are 3 sequences with 2 heads and 1 tail.
03

Identify the sequences where the first toss is heads

Among the sequences with 2 heads and 1 tail, we will identify those where the first toss is heads: 1. HHT (Heads, Heads, Tails) 2. HTH (Heads, Tails, Heads) There are 2 favorable sequences where the first toss is heads.
04

Calculate the probability

Now, we will calculate the probability by dividing the number of favorable sequences by the total number of sequences with 2 heads and 1 tail: Probability = \(\frac{Number\:of\:favorable\:sequences}{Total\:number\:of\:sequences\:with\:2\:heads\:and\:1\:tail}\) Probability = \(\frac{2}{3}\) So the probability that the first toss was heads, given that 2 heads and 1 tail fall, is \(\frac{2}{3}\).

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Most popular questions from this chapter

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