# Mathematics Questions

## Question

1. During an experiment, a card is selected at random from a standard deck of 52 cards. Event A is getting a red card. Which of the following events are independent of event A?

a) The card selected is replaced, the cards are shuffled and a second selection is made. Event B is getting a red card on this second selection.
b) The card selected is NOT replaced, the cards are shuffled and a second selection is made. Event C is getting a red card on this second selection.
c) Event D is getting a heart on the first selection.

2. A die is tossed six times in succession. Event A is getting exactly three 1’s in the six tosses. Which of the following are compatible events to event A? Explain why or why not?

a) Getting a 4 on the first die toss.
b) Rolling a 6 four times.
c) Getting three 1’s on the first three tosses.

3. On his high school’s football team, Jeffrey makes the extra point made 60% of the time. In a particular game versus the team’s biggest rivals, he had 6 extra point attempts. Assume independence between kicks.

a. Based on his average, what was the probability that all 6 attempts are successful?

b. What is the probability that at least 1 was successful?

c. What is the probability that at least 1 is not successful?

4. The probability of winning on a single game at a slot machine is 15/1,000. If you play 10 games, what is the probability that you won’t win any of the games (that is, you will lose all 10 games). You may assume independence.

5. For the 2014 season, Rickie Fowler’s putting average from within 5, 10, and 15 feet of the hole is 95%, 60%, and 35%, respectively. If Rickie participates in a putting event for charity where he has to make 3 putts from 5, 10 and 15 feet (a total of 3 putts altogether), what is his probability of doing so? Assume independence.

6. In NewYork, snow is reported 25% of days in February. If this trend continues, what is the probability that it will snow exactly 9 days this coming February (assume it is not a leap year)?

Solve this problem by using
a) The approximation mentioned in Theorem 6
b) The Binomial Distribution
and c) compare answers for a) and b)

7. 80% of disk drives made from Comdrive function properly off of the production line. During quality assurance checks, the nonfunctioning drives are removed and recycled. In a typical day, 5000 disk drives come off the production line. What is the probability that quality assurance will remove between 975 and 1050 drives?   Please choose the appropriate method to approximate this quantity.

8. The probability that I am late for work on any given day of the week is 10%.

a) Find the probability distribution of the number of times that I get to work on time over the course of one work week (5 days).
b) Company policy mandates that a manager will receive notice of an employee’s tardiness if the employee is late 2 or more days in a given week. What is the probability that my manager will be notified of my tardiness in a given week?

9. ALS is fairly rare – it’s prevalence is estimated to be about 5 in every 100,000. In the city of Danvers, there are about 30,000 people. What is the probability that there is exactly 1 person in the city of Danvers with ALS? Please use the appropriate method to approximate this quantity. (Note: you may assume independence).

10. Of the 50 ice cream flavors at J.P. Lick’s, 10 of the ice cream flavors have a vanilla base (as opposed to chocolate or some sort of other flavor base). Of the 50 ice cream flavors, 15 ice cream flavors have a candy mix-in. What is the probability that a randomly selected ice cream flavor has a vanilla base and a candy mix in?

11. A company has three different sites: Site 1, Site 2 and Site 3. At Site 1 70% of the employees are NY alum, at Site 2 20% of the employees are NY alum, at Site 3 10% are NY alum. There are an equal number of employees at each of the three sites. If an employee is randomly selected for Employee of the Month what is the probability that they are a NY alum?

## Solution Preview

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1. During an experiment, a card is selected at random from a standard deck of 52 cards. Event A is getting a red card. Which of the following events are independent of event A?

a) The card selected is replaced, the cards are shuffled and a second selection is made. Event B is getting a red card on this second selection.
This is independent of event A. The reason is that event A has no effect on the probability of event B occurring.

b) The card selected is NOT replaced, the cards are shuffled and a second selection is made. Event C is getting a red card on this second selection.
This is not independent of event A. Event C has different probabilities based on whether or not event A took place. In other words, event A has an effect on the probability of event C.

c) Event D is getting a heart on the first selection.
This is independent of event A....
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