Please show ALL work. Thank you so much! 5. Suppose you select a string of eight

Question

Please show ALL work. Thank you so much!

5. Suppose you select a string of eight bits at random. (a) What is the probability that the sum of the bits in the string is equal to 5? (b) What is the probability that the string has exactly two 0s, given that the string begins with a 0?

6. Suppose you select a string of five bits at random. Suppose event A is that the string begins with 1, event B is that the string ends with 1, and event C is that the string has exactly three 1s. (a) What is the probability of event A given event C? (b) Determine whether events B and C are independent.

7. Suppose you flip a weighted coin that has a probability of 1/4 for heads and 3/4 for tails seven times. (a) What is the probability of getting exactly six tails? (b) What is the probability of getting at least two tails?

8. Suppose a fair die has the numbers 1, 2, 2, 3, 3, and 3 on its sides. (a) If the die is rolled once (repeating multiple times), what is the expected variance of the number that appears? (b) If the die is rolled twice, what is the expected value of the product of the two numbers?

Explanation / Answer

We are allowed to do 1 question at a time. Post again for second question.

7) a) tail = 3/4 = 0.75

P(T = 6) = 7C6 * 0.75^6 * 0.25^1

P = 0.31

b) At least two tails = 1 - P(T = 0) - P(T = 1)

P = 1 - 7C0 * 0.75^0 * 0.25^7 - 7C1 * 0.75^1 * 0.25^6

P = 1 - 0.00134

P = 0.998

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