In a random experiment, two fair dice are tossed and the sum of the two dice (i.e., the sum of the numbers shown at the top faces of the dice) is recorded. (a) The player gains $3 if the sum of the two dice is less than 6 and he gains $2 if the sum of the two dice is greater than 10. Otherwise, he loses $1. Find the expected gain of the player and the standard deviation of his gain. (b) Suppose the experiments are repeated and this process stops when 8 sums greater than 6 are recorded. Find the probability that the process stops right after performing the experiments 12 times. (c) Each day, the experiments are repeated 5 times and the number of sums that are greater than 8 is recorded. The player calls that day nice if this number is at least 3. Find the probability that there are two consecutive nice days in the next week.
In a random experiment, two fair dice are tossed and the sum of the two dice (i.e., the sum of the numbers shown at the top faces of the dice) is recorded.
(a) The player gains $3 if the sum of the two dice is less than 6 and he gains $2 if the sum of the two dice is greater than 10. Otherwise, he loses $1. Find the expected gain of the player and the standard deviation of his gain.
(b) Suppose the experiments are repeated and this process stops when 8 sums greater than 6 are recorded. Find the probability that the process stops right after performing the experiments 12 times.
(c) Each day, the experiments are repeated 5 times and the number of sums that are greater than 8 is recorded. The player calls that day nice if this number is at least 3. Find the probability that there are two consecutive nice days in the next week.
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