Don't Die for Me, Argentina
An airline estimates that the probability that a random call to their reservation phone line results in a reservation being made is 0.31. This can be expressed as P(call results in reservation) = 0.31. Assume each call is independent of other calls.
What does the Law of Large Numbers say in the context of this probability?
As the number of calls becomes larger and larger, the proportion of calls resulting in a reservation will get closer and closer to 0.30.
An event A will occur with probability 0.5. An event B will occur with probability 0.6. The probability that both A and B will occur is 0.1
What is the conditional probability of A, given B?
1/6.
P(A | B) = P(A and B)/P(B) = 0.1/0.6 = 1/6.
Let the random variable X represent the weight of male black bears before they begin hibernation. Research has shown that X is approximately Normally distributed with a mean of 250 pounds and a standard deviation of 50 pounds.
What is P(X > 325) pounds?
0.0668.
P(X > 325) = P(z > 325-250/50) = P(z > 1.5) = 0.0668.
In order for the random variable X to have a geometric distribution, which of the following conditions must X satisfy?
I. p < 0.5
II. The number of trials is fixed.
III. Trials are independent.
IV. The probability of success has to be the same for each trial.
V. All outcomes in the sample space are equally likely.
I and III.
p < 0.5 and trials must be independent.
A fair coin is tossed four times, and each time the coin lands heads up.
If the coin is then tossed 1996 more times, how many heads are most likely to appear in these 1996 additional tosses?
998.
Since the first four tosses have no impact on the next 1996, the most likely result is that half of those 1996 tosses will be heads.
An airline estimates that the probability that a random call to their reservation phone line results in a reservation being made is 0.31. This can be expressed as P(call results in reservation) = 0.31. Assume each call is independent of other calls.
What is the probability that none of the next four calls results in a reservation?
0.2401.
P(four non-reservation calls) = (0.7)4 = 0.2401.
A rock concert producer has scheduled an outdoor concert. If it is warm that day, she expects to make a $20,000 profit. If it is cool, she expects to make a $5,000 profit. If it is very cold, she expects to suffer a $12,000 loss. Based upon records, the weather office has estimated the chances of a warm day to be 0.60; the chances of a cool day to be 0.25.
What is the producer's expected profit?
$11,450.
E(x) = (20,000)(0.60) + (5,000)(0.25) + (-12,000)(0.15) = $11,450.
Roll one 10-sided die 12 times. The probability of getting exactly 4 eights in those 12 rolls:
(1/10)4 x (9/10)8 = 0.000043
Use the binomial probability formula!
pk(1-p)n-k
You draw two marbles at random from a jar that has 20 red marbles and 30 black marbles without replacement.
What is the probability that both marbles are red?
0.1551.
P(Red intersect Red) = 20/50 · 19/49 = 0.1551.
If you buy a one ticket in the Provincial Lottery, then the probability that you will win a prize is 0.11. Given the nature of lotteries, the probability of winning is independent from month to month.
If you buy one ticket each month for five months, what is the probability that you will win at least one prize?
0.44.
P(at least one prize) = 1 - P(no prizes in 5 tickets) = 1 - (0.89)5 = 0.44.
A factory makes silicon chips for use in computers. It is known that about 90% of the chips meet specifications. Every hour a sample of 18 chips is selected at random for testing and the number of chips that meet specifications is recorded.
What is the approximate mean and standard deviation of the number of chips meeting specifications?
Mean = 16.2
Standard deviation = 1.273
Witney Pete, a professional dart player, has a 70% chance of hitting the bull's eye on a dartboard with any throw. Assume that each throw of a dart is independent.
Suppose Pete throws darts until he hits his first bull's eye. Find the probability that his first bull's eye occurs on the third throw.
0.063.
Geometric probability: (0.3)2(0.7) = 0.063.
A die is loaded so that the number 6 comes up three times as often as any other number.
What is the probability of rolling a 1 or a 6?
1/2.
P(6) = 3/8, P(1) = 1/8, since they are mutually exclusive, P(6 union 1) = 3/8 + 1/8 = 1/2.
A grocery store examines its shoppers' product selection and calculates the following: The probability that a randomly-chosen shopper buys apples is 0.21, that the shopper buys potato chips is 0.36, and that the shopper buys both apples and potato chips is 0.09.
Let A = randomly-chosen shopper buys apples, and C = randomly-chosen shopper buys potato chips. Sketch a Venn Diagram.
Calculate the probability that a randomly-selected shopper doesn't buy apples and doesn't buy potato chips.
0.52.
P(Ac + Cc) = 0.52.
The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Choose two pregnancies independently and at random.
What is the standard deviation of the difference in the lengths of the two pregnancies?
22.63.
Take the square root of the sum of the square of the variances, 162 + 162.
The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Choose two pregnancies independently and at random.
Find the probability that the difference in the lengths of the two pregnancies is greater than 25 days.
0.3789.
P(XD > 6 or XD < -6) = 2 · P(z > 0.88) = 0.3789.
If the heights of Icelandic women are Normally distributed with a mean of 76 inches and a standard deviation of 4.5 inches, what's the probability that Bjork is taller than you?
1 minus the p-value from Table A corresponding to the individual's z-score based on the distribution of Icelandic women.
Wile E. Coyote is pursuing the Road Runner. The Road Runner chooses his route randomly, such that there is a probability 0.8 that he'll take the high road, and 0.2 that he'll take the low road. If he takes the high road, the probability that Wile E. catches him is 0.01. If he takes the low road, the probability he gets caught is 0.05.
Find the probability that he took the high road, given that he was caught.
0.444.
P(HR | Caught) = P(HR union Caught)/P(Caught) = (0.8)(0.01)/(0.8)(0.01)+(0.2)(0.05) = 0.44
Picard Partners is planning a major investment. The expected value (the mean) of the profit (in millions of dollars) is estimated to be around 3. If this kind of investment was made many times, the expected average distance between the amount of profit and the mean profit (the standard deviation) is around 2.53.
Picard owes its source of capital a fee of $200,000 plus 10% of the profits X. So the firm actually retains Y = 0.9X - 0.2 from the investment.
Use a linear transformation to find the mean and standard deviation for Y.
μY = $2.50 and σY = $2.28.
μY = 0.9(3) - 0.2 = $2.50, and σY = 0.9(2.53) = $2.28.
Witney Pete, a professional dart player, has a 70% chance of hitting the bull's eye on a dartboard with any throw. Assume that each throw of a dart is independent.
What is the probability that Pete hits 5 or fewer of his next 10 shots?
0.0130.
P(X ≤ 5) = binomcdf (10, 0.84, 5) = 0.0130.