Respuesta :
The expected number of cards Henry needs to turn up to get the third ace is 31.8
Given: We need to find what is the expected number of cards Harry needs to turn up to get the third ace and Harry has a standard deck of 52 cards.
Now, we know there are 4 ace cards. Let’s consider the ace cards and non-ace cards separately.
There are (52-4) = 48 non-ace cards.
Let us consider that the non-ace cards will be cut by an ace card.
So, there are 5 possible ways in which the ace cards can cut the division, provided
We are having an equal number of non-ace cards in between the appearance of each ace card which means the setup is symmetric among the non-ace cards.
So let us name the spaces between ace cards as s1, s2, s3, s4, and s5.
Therefore, the position of the third ace card is equal to s1 + s2 + s3 + 3.
The expected value of this position is E[s1 + s2 + s3 + 3].
By linearity of expectation, E[s1 + s2 + s3 + s4] is E[s1] + E[s2] + E[s3] + 3.
As the setup is symmetric between the five places, E[s1] = E[s2] = E[s3] = E[s4] = E[s5]
And since E[s1 + s2 + s3 + s4 + s5] = E[s1] + E[s2] + E[s3] + E[s4] + E[s5] = 48
Therefore, E[s1] = E[s2] = E[s3] = E[s4] = E[s5] = 48 / 5
The expected value of position is E[s1] + E[s2] + E[s3] + 3
= 48 / 5 + 48 / 5 + 48 / 5 + 3
= 3 * 48 / 5 + 3
= 31.8
Hence, the expected number of cards Henry needs to turn up to get the third ace is 31.8
Know more about “statistical probability” here: https://brainly.com/question/15006619
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