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Chapter 2: Q22SE (page 91)

Suppose that a balanced die is rolled repeatedly until the same

number appears on two successive rolls, and let X denote the number of

rolls that are required. Determine the value of Pr(X= x), for x=2, 3,...

Short Answer

Expert verified

The probability is,

\(P\left( {X = x} \right) = {\left( {\frac{5}{6}} \right)^{x - 2}}\left( {\frac{1}{6}} \right)\,;\,for\,x = 2,3,...\)

Step by step solution

01

Given information

A balanced die is rolled repeatedly until the same number appears on two

Successive rolls.

The X denotes the number of rolls that are required.

02

To compute the value of P(X=x), for x=2,3, …

We require the same number to appear on the two successive rolls.

Let us assume that the (x-1)roll is 6. Then we need the xthroll to be 6 as well,

which occurs with a probability of 1/6.

The xth roll is the same as the previous roll.

Therefore,

The required probability is,

\(P\left( {X = x} \right) = {\left( {\frac{5}{6}} \right)^{x - 2}}\left( {\frac{1}{6}} \right)\,\,;\,for\,x = 2,3,...\)

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Most popular questions from this chapter

Consider a machine that produces items in sequence. Under normal operating conditions, the items are independent with a probability of 0.01 of being defective. However, it is possible for the machine to develop a 鈥渕emory鈥� in the following sense: After each defective item, and independent of anything that happened earlier, the probability that the next item is defective is\(\frac{{\bf{2}}}{{\bf{5}}}\). After each non-defective item, and independent of anything that happened earlier, the probability that the next item is defective is\(\frac{{\bf{1}}}{{{\bf{165}}}}\).

Please assume that the machine is operating normally for the whole time we observe or has a memory for the whole time we observe. LetBbe the event that the machine is operating normally, and assume that\({\bf{Pr}}\left( {\bf{B}} \right){\bf{ = }}\frac{{\bf{2}}}{{\bf{3}}}\). Let\({{\bf{D}}_{\bf{i}}}\)be the event that theith item inspected is defective. Assume that\({{\bf{D}}_{\bf{1}}}\)is independent ofB.

a. Prove that\({\bf{Pr}}\left( {{{\bf{D}}_{\bf{i}}}} \right){\bf{ = 0}}{\bf{.01}}\) for alli. Hint:Use induction.

b. Assume that we observe the first six items and the event that occurs is\({\bf{E = D}}_{\bf{1}}^{\bf{c}} \cap {\bf{D}}_{\bf{2}}^{\bf{c}} \cap {{\bf{D}}_3} \cap {{\bf{D}}_4} \cap {\bf{D}}_{\bf{5}}^{\bf{c}} \cap {\bf{D}}_{\bf{6}}^{\bf{c}}\). The third and fourth items are defective, but the other four are not. Compute\({\bf{Pr}}\left( {{\bf{B}}\left| {\bf{E}} \right.} \right)\).

Suppose that in Example 2.3.4 in this section, the item selected at random from the entire lot is found to be non-defective. Determine the posterior probability that it was produced by machine\({{\bf{M}}_{\bf{2}}}\).

Suppose that the events A and B are disjoint and that

each has positive probability. Are A and B independent?

If three balanced dice are rolled, what is the probability that all three numbers will be the same?

Consider again the conditions of Exercise 2 of Sec. 1.10. If a family selected at random from the city subscribes to newspaper A, what is the probability that the family also subscribes to newspaper B?
See all solutions

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