# Probability: Word Problems

1. When transmitting messages from a point A to a point B, out of every 40 messages 6 need to be corrected by applying error correcting codes. What is the probability that in a batch of 200 messages sent from A to B, there will be between 38 and 42 messages that will have to be corrected.

2. The probability of an event A occurring in each of a series of independent trials is 2/3. Find the distribution function of the number of occurrences of A in 9 trials.

3. The probability that a network will be shut down on any given day is .0003. What is the probability that the network will be shut down twice in 5000 days.

4. 82% of items are "regular" and 77% of regular items are "top quality." What is the probability that a randomly picked regular item is of the top quality?

5. You have two boxes. 27 out of 30 items in the first box are "regular" and 11 out of 15 items in the second box are "regular." A randomly chosen item is taken from the second box and placed in the first. What is the probability that a randomly picked item from the first box is regular?

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#### Solution Preview

1.

The number of messages that need to be corrected out of 200, X, follows a Normal distribution with mean 200(6/40) = 30 and variance 200(6/40)(1-6/40) = 25.5. We then need to find out

Pr(38-0.5 < X < 42+0.5)

Where we subtract and add 0.5 in the appropriate places for a continuity adjustment to the normal distribution. This calculation then becomes:

Pr( (37.5 - 30) / sqrt(25.5) < (X - 30) / sqrt(25.5) < ...

#### Solution Summary

Five probability word problems are examined.

Algebra - Word Problem

Write a system of two equations in two unknowns for each problem.

1. Investing her bonus. Donna invested her $33,000 bonus and received a total of $970 in interest after one year. If part of the money returned 4% and the remainder 2.25%, then how much did she invest at each rate?

2. Ticket sales. Tickets for a concert were sold to adults for $3 and to students for $2. If the total receipts were $824 and twice as many adult tickets as student tickets were sold, then how many of each were sold?

3. Textbook case. The accompanying graph shows the cost of producing textbooks and the revenue from the sale of those textbooks.

a) What is the cost of producing 10,000 textbooks?

b) What is the revenue when 10,000 textbooks are sold?

c) For what number of textbooks is the cost equal to the revenue?

d) The cost of producing zero textbooks is called the fixed cost. Find the fixed cost.

4. Free market. The function S = 5000 + 200x and D = 9500 - 100x express the supply S and the demand D, respectively, for a popular compact disc brand as a function of its price x (in dollars).

a) Graph the functions on the same coordinate system.

b) What happens to the supply as the price increases?

c) What happens to the demand as the price increases?

d) The price at which supply and demand are equal is called the equilibrium price. What is the equilibrium price?

5. Solve for x and y.

2y-x =3

x=3y-5

6. Books and magazines. At Gwen's garage sale, all books were one price, and all magazines were another price. Harriet bought four books and three magazines for $1.45, and June bought two books and five magazines for $1.25. What was the price of a book and what was the price of a magazine?

7. Low-fat yogurt. Ziggy's Famous Yogurt blends regular yogurt that is 3% fat with its no-fat yogurt to obtain lowfat yogurt that is 1% fat. How many pounds of regular yogurt and how many pounds of no-fat yogurt should be mixed to obtain 60 pounds of low-fat yogurt?

8. Super Bowl contender. The probability that San Francisco plays in the next Super Bowl is nine times the probability that they do not play in the next Super Bowl. The probability that San Francisco plays in the next Super Bowl plus the probability that they do not play is 1. What is the probability that San Francisco plays in the next Super Bowl?

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