Question 1203779
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The logical way to answer these questions is to do part (e) first and -- to verify that your calculations are correct -- verify that the sum of the probabilities is 1; then answer the individual questions.<br>
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NOTE!

If you are just learning how to calculate binomial probabilities, it is good practice -- if the number of cases is small -- to calculate the probabilities of all possible outcomes and verify that their sum is 1

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3 of 3 alive at age 65:
{{{C(3,3)((.8)^3)((.2)^0)=(1)(.512)(1)=.512}}}<br>
2 of 3 alive at age 65:
{{{C(3,2)((.8)^2)((.2)^1)=(3)(.64)(.2)=.384}}}<br>
1 of 3 alive at age 65:
{{{C(3,1)((.8)^1)((.2)^2)=(3)(.8)(.04)=.096}}}<br>
0 of 3 alive at age 65:
{{{C(3,0)((.8)^0)((.2)^3)=(1)(1)(.008)=.008}}}<br>
.512+.384+.096+.008 = 1.000 so the calculations should be good.<br>
ANSWERS:
(a) exactly 2: 0.384
(b) at least 1: 1-.008 = 0.992
(c) at most 2: 1-.512 = 0.488
(d) less than 1: 0.008
(e) (see above)<br>