(Business Mathematics &Statistics)?

Question

The probability that a man aged 50 years will die within a year is 0.01125.What is the probability that of 12 such persons at least 11 will reach their fifty-first birthday?

Answer 1

The number of people aged 50 out of 12 people who will reach 51 has a binomial distribution. That is because there is a set number of trials (12), the probability each one reaches 51 is the same for each person (1-0.01125), and the people are independent of each other.

P(At least 11 will reach 51) = P(11 will reach 51 OR 12 will reach 51)
= P(11 will reach 51) + P(12 will reach 51) since these are mutually exclusive events.
= (12!/11!1!) * (1-0.01125)^11 * (0.01125^1) +
(12!/12!0!) * (1-0.01125)^12 * (0.01125)^0
= 0.0121 + 0.0824
= 0.0945

Answer 2

P(n(51)>=11) = probability that the number of old geezers is greater than or equal to 11 people.

p(croak) = probability that a 50 yo man will croak in a year.

p(NOT croak) = 1 - p(croak).

Therefore the probability that at least eleven of these wheezers survives their 51st bday is the joint probability that eleven do not croak. That is, at least 11 have to survive given a probability of surviving for each one at p(NOT croak) = 1 - p(croak); where p(croak) = .01125

In math talk P(n(51)>=11) = p(NOT croak) X p(NOT croak) X ... X p(NOT croak) for eleven guys, each having p(NOT croak) probability of surviving the year.

Thus, P(n(51)>= 11) = p(NOT croak)^11 = (1 - .01125)^11; and you can do the math.

Lesson learned: Joint probability of independent events (like a man dying or surviving) is just the multiple of each event. In this problem, we needed to know the joint probability of at least 11 geezers surviving to their 51st b'day. So we found the probability of not croaking (i.e., of surviving) for each one and multiplied that eleven times to indicate the joint probability that at least eleven survive.

You can test the logic of a joint probability in your own mind by asking this question "Will the probability of at least some number surviving go up or down if we increase the number who must survive?" If you find your joint probability is going up with increased numbers, you've probably done something wrong in your modeling. That is, it has to be more difficult to get more survivors, for example, than fewer and more difficult translates to lower probability.

Answer 3

P(n(51)>=11) = danger that the style of previous geezers is larger than or equivalent to 11 human beings. p(croak) = danger that a 50 yo guy will croak in a 365 days. p(no longer croak) = a million - p(croak). hence the risk that a minimum of 11 of those wheezers survives their 51st bday is the joint danger that 11 do no longer croak. it particularly is, a minimum of 11 ought to proceed to exist given a danger of surviving for each physique at p(no longer croak) = a million - p(croak); the place p(croak) = .01125 In math communicate P(n(51)>=11) = p(no longer croak) X p(no longer croak) X ... X p(no longer croak) for 11 adult males, each and every having p(no longer croak) danger of surviving the 365 days. to that end, P(n(51)>= 11) = p(no longer croak)^11 = (a million - .01125)^11; and you will do the maths. Lesson discovered: Joint danger of self reliant activities (like a guy death or surviving) is purely the dissimilar of each and every experience. in this concern, we mandatory to renowned the joint danger of a minimum of 11 geezers surviving to their 51st b'day. So we got here upon the risk of no longer croaking (i.e., of surviving) for each physique and accelerated that 11 circumstances to show the joint danger that a minimum of 11 proceed to exist. you may attempt the logic of a joint danger on your guy or woman strategies by ability of asking this question "Will the risk of a minimum of a few selection surviving pass up or down if we boost the selection who ought to proceed to exist?" in case you detect your joint danger is going up with greater advantageous numbers, you have in all hazard performed some thing incorrect on your modeling. it particularly is, it must be greater difficult to get greater survivors, working example, than fewer and greater difficult interprets to decrease danger.

Answer 4

11*(1-.01125)^11*.01125 +(1-.01125)^12=.982