Question

1.Two dice are tossed 432 times. How many times would you expect to get a sum of 5?

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2.Sam is applying for a single year life insurance policy worth $35,750.00. If the actuarial tables determine that she will survive the next year with probability 0.996, what is her expected value for the life insurance policy if the premium is $433.00 ?

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3.A raffle is being held at a benefit concert. The prizes are awarded as follows: 1 grand prize of

$6,200.00, 3 prizes of $1,000.00, 4 prize of $92.00, and 12 prizes of $25.00.

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4.Find the expected value for the random variable:

X 1 3 4 6

P(X) 0.21 0.12 0.23 0.44

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5.Suppose that from a standard deck, you draw three cards without replacement. What is the expected number of aces that you will draw?

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6.Consider 3 trials, each having the same probability of success. Let

X

X

denote the total number of successes in these trials. If E[X]=0.6, find each of the following.

(a) The largest possible value of P{X=3}:

P{X=3}≤

(b) The smallest possible value of P{X=3}:

P{X=3}≥

In this case, give possible values for the remaining probabilities:

P{X=0}=

P{X=1}=

P{X=2}=

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7.It is reasonable to model the number of winter storms in a season as with a Poisson random variable. Suppose that in a good year the average number of storms is 5, and that in a bad year the average is 8. If the probability that next year will be a good year is 0.3 and the probability that it will be bad is 0.7, find the expected value and variance in the number of storms that will occur.

expected value =

variance =

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8.In a popular tale of wizards and witches, a group of them finds themselves in a room with doors which change position, making it impossible to determine which door is which when the room is entered or reentered. Suppose that there are 4 doors in the room. One door leads out of the building after 3 hours of travel. The second and third doors return to the room after 5 and 5.5 hours of travel, respectively. The fourth door leads to a dead end, the end of which is a 2.5 hour trip from the door.

If the probabilities with which the group selects the four doors are 0.2, 0.1, 0.1, and 0.6, respectively, what is the expected number of hours before the group exits the building?

E[Number of hours]=

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9.For a group of 100 people, assuming that each person is equally likely to have a birthday on each of 365 days in the year, compute

(a) The expected number of days of the year that are birthdays of exactly 4 people:

E[days with 4 birthdays]=

(b) The expected number of distinct birthdays:

E[distinct birthdays]=

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10.Consider 35 independent flips of a coin having probability 0.5 of landing on heads. We say that a changeover occurs when an outcome is different from the one preceding it. Find the expected number of changeovers.

E[changeovers]=

Answer 1

#1) If two dice are tossed then there are 4 outcomes of getting sum 5 { (1,4),(2,3),(3,2),(4,1) } out of total 36 outcomes.

So probability of getting sum 5 is 4/36

If two dice are tossed 432 times, so expected number to get sum 5 = 432*4/36 = 48

#2) Let x be the amount of loss or gain for Sam P(x) would be the corresponding probability.

We are given that probability that she will survive the next year is 0.996 , so she would be occurred loss of premium -$433

If she will not survive , she would be occurred gain of 35750 - 433 = $35317

So expected value =

.

So expected value = -$290

#3) Question is incomplete.

#4) expected value =

Expected value = 4.13

#5) In a standard deck there are total 52 cards out of 4 are ace

Let x be the number of aces

If x = 0 that is no ace card  ,then P(x =0 ) = = 17296 / 22100 = 0.7826

If x = 1 that is one ace card and two non ace card  ,then P(x =1) = = 4*1128 / 22100 = 0.2042

If x = 2 that is two ace cards and one non ace card  ,then P(x =1) = = 6*48/ 22100 = 0.0130

If x = 3 that is all three are ace cards ,then P(x = 3 ) = = 4 / 22100 = 0.0002

Therefore expected number of aces are 0.2308