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Question 14, Page 278

Mandy is 30, and the probability that she will survive until age 65 is 0.90. Ashley is 45, and the probability that she will survive until age 65 is 0.95.

(a) Find the probability that both Mandy and Ashley will survive until age 65.
(b) Find the probability that only Mandy will survive until age 65.
(c) Find the probability that neither Mandy nor Ashley will survive until age 65.
(d) What assumption about the lives of Mandy and Ashley did you make in answering the above questions?

Source: Discovering Statistics by James Hawkes, Second Edition, Exercise Set B, Section 6.5

Sagot :

GinnyAnsweridn
Certainly! Let's address each part of the question step by step.

### (a) Probability that both Mandy and Ashley will survive until age 65

Given:
- The probability that Mandy will survive until age 65 is 0.90.
- The probability that Ashley will survive until age 65 is 0.95.

To find the probability that both Mandy and Ashley will survive until age 65, we multiply their individual probabilities, assuming their survival rates are independent.

[tex]\[ \text{Probability both will survive} = P(Mandy \text{ survives}) \times P(Ashley \text{ survives}) \][/tex]
[tex]\[ = 0.90 \times 0.95 \][/tex]
[tex]\[ = 0.855 \][/tex]

So, the probability that both Mandy and Ashley will survive until age 65 is 0.855.

### (b) Probability that only Mandy will survive until age 65

Given the same probabilities:
- The probability that Mandy will survive until age 65 is 0.90.
- The probability that Ashley will not survive until age 65 is [tex]\(1 - 0.95 = 0.05\)[/tex].

To find the probability that only Mandy will survive until age 65, we multiply Mandy's survival probability by the probability that Ashley does not survive.

[tex]\[ \text{Probability only Mandy will survive} = P(Mandy \text{ survives}) \times P(Ashley \text{ does not survive}) \][/tex]
[tex]\[ = 0.90 \times 0.05 \][/tex]
[tex]\[ = 0.045 \][/tex]

So, the probability that only Mandy will survive until age 65 is 0.045.

### (c) Probability that neither Mandy nor Ashley will survive until age 65

To find the probability that neither Mandy nor Ashley will survive until age 65:

- The probability that Mandy will not survive until age 65 is [tex]\(1 - 0.90 = 0.10\)[/tex].
- The probability that Ashley will not survive until age 65 is [tex]\(1 - 0.95 = 0.05\)[/tex].

We multiply these probabilities:

[tex]\[ \text{Probability neither will survive} = P(Mandy \text{ does not survive}) \times P(Ashley \text{ does not survive}) \][/tex]
[tex]\[ = 0.10 \times 0.05 \][/tex]
[tex]\[ = 0.005 \][/tex]

So, the probability that neither Mandy nor Ashley will survive until age 65 is 0.005.

### (d) Assumption

In answering the above questions, the primary assumption made is that the survival times of Mandy and Ashley are independent of each other. This means that the probability of Mandy surviving until age 65 does not affect the probability of Ashley surviving until age 65, and vice versa.
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