# Probability Review

Two coins are tossed simultaneously. What is the probability of tossing two heads?

A coin is tossed three times. What is the probability of tossing exactly two heads?

Two standard dice are rolled. What is the probability of rolling a total of 9?

If the odds in favour of rain tomorrow are 4:7, what is the probability of rain tomorrow?

If a bowl contains ten chocolates and eight jelly beans, what is the probability that four of the candies will be chocolates?

Two standard dice are rolled. What is the probability that a sum less than 7 is not rolled?

A stable has 15 horses available for trail rides. Of these horses, 6 are all brown, 5 are mainly white and the rest are all black. If Maria selects one at random, what is the probability that this horse will: a) be black? b) not be black?

A number is chosen randomly from the first 20 natural numbers (1 to 20). If event A = {a multiply of 5}, what is the value of P(A')?

If the odds against the Blue Jays winning this year's World Series are 20:1, what is the probability that the Blue Jays will win this series?

Describe how to calculate the odds against an event happening when you know the probability of the event occuring. Use a numerical example to illustrate your explanation.

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#### Solution Preview

1) Two coins are tossed simultaneously. What is the probability of tossing two heads?

The sample space consists of four equally probable outcomes: HH, HT, TH, TT. The favorable event contains only one outcome, HH. So the probability of tossing two heads is 1/4

2) A coin is tossed three times. What is the probability of tossing exactly two heads?

The sample space now has 8 possible outcomes:

HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Of these oucomes, HHT, HTH and THH have exactly two heads. So, the favorable event consists of 3 oucomes out of 8, and its probability is 3/8.

3) Two standard dice are rolled. What is the probability rolling a total of 9?

Let's construct the sample space. A table is usually helpful.

| 1 2 3 4 5 6

-------------------------------------------------

1 | 2 3 4 5 6 7

2 | 3 4 5 6 7 8

3 | 4 5 6 7 8 9

4 | 5 6 7 8 9 10

5 | 6 7 8 9 10 11

6 | 7 8 9 10 11 12

The leftmost column represents the numer rolled on the first dice, the topmost row contains the number rolled on the second dice.

The table contains the corresponding sums of numbers. The event that we want, E={the sum is 9}, contains ...

#### Solution Summary

Describe how to calculate the odds against an event happening when you know the probability of the event occuring. Use a numerical example to illustrate your explanation.

Probability Review Questions

Question 1 The following five (5) situations require the application of one of the common discrete or common continuous distributions we learned about. All you have to do for each of these problems is identify the correct probability mass function or probability density function name. DO NOT SOLVE THE PROBLEM. There is only one correct answer for each problem and each answer may (or may not) be used more than once. Your choices of possible answers include the following:

Bernoulli pmf Binomial pmf Poisson pmf Exponential pdf Uniform pdf

1a. Ten independent rocket missions to the Mars are planned by NASA. The probability that a rocket makes it to Mars successfully is .95. What is the probability that at least 8 of these rockets make it to Mars? What pmf or pdf would you use to solve this problem ?

1b. A vehicle is randomly located on a 400-mile stretch of I-10 between Phoenix and Los Angeles. The location of this vehicle, X, has CDF F(x) = 1/400 for 0 < x < 400. What pmf or pdf does X have?

1c. Pedestrians cross at an intersection. T, their inter-arrival times (times between consecutive arrivals) has this CDF: F(t) = 1 - e-t for t > 0. What pdf or pmf does T have?

1d. A certain genetic test can determine if a person is a carrier of breast cancer gene or not. X = 1 with probability p if a person is a carrier and X = 0 with probability 1-p if a person is not a carrier. What pmf or pdf does X have?

1e. An intersection has 30 cars/min pass through it. The probability that x = 25 pass through this intersection in one minute is

What is the name of the pdf or pmf of X?

Question 2. The weight of small motor is modeled as a uniformly distributed random variable between 10 and 11 lbs. For ease, call this weight W.

a. Graph the pdf of W. Label your axes and show any important numerical values on them.

b. Graph the CDF of W. Label your axes and show any important numerical values on them.

c. Twenty (20) of these motors will be placed into a shipping box. If the weight of a box exceeds 212 lbs, there will be an extra shipping charge of $20.00 applied. What is the probability that the shipping charge is applied?

Bonus : What is the probability that the weight of a randomly selected box is 11 lbs?

Question 3 Bob is responsible for inspecting Product A and Product B. Both products arrive at his inspection station independently of one another. Product A arrives to his station at a rate of 3 every 30 minutes according to a Poisson pmf. Product B arrives to his station at a rate of 5 every 30 minutes also according to a Poisson pmf.

a. What is the mean number of Product A that Bob inspects in one hour?

b. What is the probability that Bob inspects more than 8 of Product B in the next hour? You can leave your answer in term (summation) form. No need to crunch the number.

Question 4 At a restaurant, daily demand (D) for a specific dish (e.g., Peking Duck) has the following associated probabilities. (There has never been more than three requests for this dish):

P(D = 0) = .10 P(D = 1) = .40 P(D = 2) = .30 P(D = 3) = .20

a. What is the probability that the demand is 4?

b. What is the expected demand?

c. What is the variance of the demand?

Question 5 Random variable X has p.d.f. for 2 < x < m

a. What is the value of m?

b. What is P(X =2) + P(X = 5)?

c. What is the probability that X is larger than 5?

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