Conditional Probability

Show Menu

Probability

THE M&M CANDYCHINE

Click the Blue Lever

PROBABILITIES OF SUCCESSIVE EVENTS

What would be the probability of being dealt three cards and all three of them turning out to be aces? One way to calculate the probability of being dealt three cards all of which are aces is to count the number of ways of getting three of the four aces in the deck:

₄C₃ = 4 x 3 x 2 / 3! = 4

and the number of ways of getting 3 cards out of the 52 cards in the deck is

₅₂C₃ = 52 x 51 x 50 = 22,100

Dividing the numbers of ways, we find that the probability is

4/22,100 = 1/5,525

Ace of Diamonds

Another way to figure out this probability is to reason as follows. The probability that the first card is an ace is 4/52 because 4 of the 52 cards in the original deck are aces. After the first ace has been dealt, the probability that the second card is an ace is 3/51 because there are 3 aces left out of 51 cards. After the second ace has been dealt, the probability that the third card is an ace is 2/50 because there are 2 aces left out of 50 cards. If these three probabilities are multiplied, we get

4/52 x 3/51 x 2/50 = 24/132,600 = 1/5,525

The fact that this result agrees with the previous one suggests that probabilities can be multiplied in the same way that numbers of choices are multiplied.

To find the probability of several events occurring in succession, multiply the probabilities of the individual events.

In figuring out the probability of being dealt three aces, the probability of each successive event is affected by the occurrence of the previous one.

When events affect each other, their probabilities are called conditional.

Events that are influenced by other events are said to be dependent.

Suppose, instead, that you are playing a card game in which the deck is shuffled and dealt several times during the game. What is the probability that, on three successive deals, the first card you are dealt is an ace? In this case, the probability of each successive event is not affected by the occurrence of the previous one.

Events that have no influence on each other are said to be independent.

On each deal, the probability of getting an ace is the same because each deal starts with all 4 aces in a complete deck of 52 cards. The probability is

4/52 x 4/52 x 4/52 = 64/140,608 = 1/ 2,197

The fact that this result is larger than the previous one shows that being dealt an ace at the beginning of three successive deals is more likely than getting three aces all in one deal.

PART 1: QUESTIONS

Suppose that a coin is tossed 2 times in succession.
1. What is the probability that it turns up tails on the first toss?
2. What is the probability that it turns up tails on the second toss?
3. What is the probability that it turns up tails on both tosses?
4. Are the events of the coin turning up tails on the first toss and on the second toss dependent or independent?
Suppose that a bag contains three coins: a penny, a nickel, and a dime.
1. What is the probability that if a coin is chosen at random, it will be the penny?
2. Suppose a coin is chosen and it turns out to be the penny.
3. What is the probability that if a second coin is chosen, it will turn out to be the nickel?
4. What is the probability that the first coin will be the penny and the second coin will be the nickel?
5. Are the events of getting a penny and then a nickel dependent or independent?
The probability that a high school athlete in baseball, basketball, or football goes on to play at the college level is 1/20. The probability that one of these athletes goes from college to professional sports is 17/1000.
1. What is the probability that a high school athlete goes on play in both college and professional sports?
2. Out of 20,000 high school athletes, about how many go on to both college and professional sports?
In the United States, the probability that a man will live to the age of 75 is about 3/10 and the probability that a woman will live to the age of 75 is about 1/2. On the assumption that these probabilities are true for a person's grandparents, find each of the following probabilities.
1. The probability that both grandparents on a person�s father�s side will live to the age of 75.
2. The probability that both grandparents on a person�s mother�s side will live to the age of 75.
3. The probability that all four of a person�s grandparents will live to the age of 75.
4. Do you think it is correct to assume that all four events are independent?
The actor Sean Connery once bet on the number 17 three times in succession in a roulette game in the St. Vincent casino. All three times the number 17 came up, and Mr. Connery won $20,000. The wheel had 37 compartments, numbered 0 and 1 through 36. What is the probability that the number 17 would come up
1. on a single spin?
2. on two successive spins?
3. on three successive spins?
A restaurant has four cooks, each of whom has a 7% probability of being absent from work.
1. Express the probability of 7% as a decimal fraction.
2. What is the probability that two of the cooks will be absent on the same day?
3. What is the probability that all four cooks will be absent at the same time?
4. Do you think it is reasonable to assume that the event of each cook being absent is always independent of the others? Explain.
The probability that one of three jet engines will fail during any one flight is .0001.
1. What is the probability that all three will fail?
2. What is the probability that all three will fail if serviced by the same mechanic?
3. In which case are the events independent?
4. In which case are the events dependent?
5. What can be done to lower the probability of engine failure?

PART 2: BLUE M & M's

Click the Blue Lever

Before you is an M&M dispenser jar containing 100 M&M's. The amounts for each color are displayed beneath the dispenser. If you push the blue lever, one M&M drops out.
What is the chance of getting ...

a blue M&M on your first try?
no blue M&M on your first try?
a red M&M the first try and then a blue M&M the second try?
a blue M&M the first try and a blue M&M the second try?
a blue M&M if you already got two blue M&Ms your first and second tries?
Careful! Tricky question...
a blue or a red or a green M&M on your first try?
no blue M&Ms in five tries?
at least one blue M&M in five tries?
all blue M&Ms in five tries?
a blue M&M if all 100 M&Ms were put back in the jar?

Make sure to show all your calculations.
Write your answer as a FRACTION, a DECIMAL, and a PERCENTAGE.
Use at least 3 digit precision.
Isolated answers are not accepted.

Comment on this Page

Last Modified 12/12/08 9:22 AM