A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ2 denote the mean and variance of X, then the value of 64(μ+σ2) is:
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Step-by-Step Solution
Step 1: List Sample Space and Values of X
First, we list all possible outcomes when a coin is tossed three times. There are 23=8 possible outcomes. For each outcome, we determine the value of X, which is the number of times a tail follows a head (HT sequence).
Step 2: Determine Probability Distribution of X
From the sample space, we count the occurrences for each value of X: X=0 (HHH), X=1 (HHT, THH, THT, TTH), X=2 (HTH, HTT), X=3 (none). Wait, let's re-evaluate the counts for X. HHH has 0 HT. HHT has 1 HT. HTH has 1 HT. THH has 0 HT. HTT has 1 HT. THT has 1 HT. TTH has 0 HT. TTT has 0 HT. So, X=0 for HHH, THH, TTH, TTT (4 outcomes). X=1 for HHT, HTH, HTT, THT (4 outcomes). X=2 and X=3 have 0 outcomes. This means the problem statement or my interpretation of 'tail follows a head' might be slightly off. Let's re-read carefully: 'number of times a tail follows a head'.
Let's re-evaluate X for each outcome:
HHH: 0 (no HT)
HHT: 1 (H_H_T)
HTH: 1 (H_T_H)
THH: 0 (no HT)
HTT: 1 (H_T_T)
THT: 1 (T_H_T)
TTH: 0 (no HT)
TTT: 0 (no HT)
So, X=0 for HHH, THH, TTH, TTT (4 outcomes).
X=1 for HHT, HTH, HTT, THT (4 outcomes).
This means P(X=0)=4/8=1/2 and P(X=1)=4/8=1/2. All other probabilities are 0. Let's correct the table.
Step 3: Calculate Mean (μ)
The mean, μ, also known as the expected value E(X), is calculated by summing the product of each possible value of X and its corresponding probability.
Step 4: Calculate Variance (σ²)
The variance, σ2, is calculated using the formula E(X2)−(E(X))2. First, we find E(X2) by summing the product of the square of each X value and its probability. Then, we subtract the square of the mean.
Step 5: Calculate the Final Expression
Finally, we substitute the calculated values of the mean (μ) and variance (σ2) into the given expression 64(μ+σ2) and simplify to find the final answer.