One coin is fair and one is weighted so that Pr[H] = [ 1/4]. You randomly select one of the two coins, and flip it 3 times, noting the result of each flip.
What is the expected number of heads?
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To find the expected number of heads, we need to consider the probabilities of getting heads with each coin and the probability of selecting each coin.
Let's start by calculating the probabilities of getting heads with each coin:
For the fair coin, Pr[H] = 1/2 since it is a fair coin.
For the weighted coin, Pr[H] = 1/4 since it is weighted and has a lower probability of landing on heads.
Now let's consider the probability of selecting each coin:
Since we randomly select one of the two coins, the probability of choosing the fair coin is 1/2, and the probability of selecting the weighted coin is also 1/2.
To find the expected number of heads, we multiply the probability of getting heads with each coin by the probability of selecting that coin, and sum up the results:
Expected Number of Heads = (Pr[H] * Pr[Select fair coin]) + (Pr[H] * Pr[Select weighted coin])
Expected Number of Heads = (1/2 * 1/2) + (1/4 * 1/2)
Expected Number of Heads = 1/4 + 1/8
Expected Number of Heads = 3/8
Therefore, the expected number of heads is 3/8.