Probability Tester

1. Suppose E is an event in a sample space S with probability .3. What is the probability of the complement of E?

1

.3

.7

0

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Explanation

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2. Two six-sided dice are rolled. Find the probability that the sum is 8 given that the first die was 3.

1/5

1/6

5/18

1/12

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Explanation

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3. The results of the study in question #9 on the relationship between a certain drug and the improvement in symptoms for patients with a particular medical conditon are repeated below.

	Improvement	No Improvement	Total
Drug	270	530	800
No Drug	120	280	400
Total	390	810	1200

Based on this table, are the events "has the drug" and "improvement" independent events?

No, they are not independent

Yes, they are independent

The table shows the number of patients who have taken the drug and whether they experienced improvement or not. To determine if the events "has the drug" and "improvement" are independent, we need to compare the observed frequencies with the expected frequencies under independence. If the observed frequencies are significantly different from the expected frequencies, then the events are not independent. In this case, the observed frequencies are not equal to the expected frequencies, indicating that the events are not independent. Therefore, the correct answer is "No, they are not independent."

Explanation

The table shows the number of patients who have taken the drug and whether they experienced improvement or not. To determine if the events "has the drug" and "improvement" are independent, we need to compare the observed frequencies with the expected frequencies under independence. If the observed frequencies are significantly different from the expected frequencies, then the events are not independent. In this case, the observed frequencies are not equal to the expected frequencies, indicating that the events are not independent. Therefore, the correct answer is "No, they are not independent."

4. Two six-sided dice are rolled. But this time, the dice aren't fair: For each die, a 1 is twice as likely to be rolled as a 2, a 2 is twice as likely to be rolled as a 3, ..., and a 5 is twice as likely to be rolled as a 6 (in other words, each number is twice as likely as the number that follows it). So what is the probability of rolling a sum of 7?

64/1023

64/1223

64/1323

64/1123

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Explanation

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5. In a group of 339 people, at least two of them have the same first-name and last-name initials, possibly switched (as in Constance Smith and Selwyn Crown)

True

False

In a group of 339 people, it is not guaranteed that at least two of them will have the same first-name and last-name initials, possibly switched. It is possible that all 339 people have unique initials or that some initials are repeated but not switched. Therefore, the statement is false.

Explanation

In a group of 339 people, it is not guaranteed that at least two of them will have the same first-name and last-name initials, possibly switched. It is possible that all 339 people have unique initials or that some initials are repeated but not switched. Therefore, the statement is false.

6. We're playing poker with a standard pack of fifty-two cards without jokers. You have two pairs: 3, 3, 4, 4, 8. You are unaware of which cards the other players are holding. You decide to replace the 8 by another card from the pack. How do you rate your chances to turn your hand into a full house? (Full house means one trio and one pair.)

1/26

1/13

4/47

2/13

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Explanation

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7. A company has three plants at which it produces a certain item. 30% are produced at Plant A, 50% at Plant B, and 20% at Plant C. Suppose that 1%, 4% and 3% of the items produced at Plants A, B and C respectively are defective. If an item is selected at random from all those produced, what is the probability that the item was produced at Plant B and is defective?

.02

None of the answers given

.2

.04

The probability that an item was produced at Plant B and is defective can be calculated by multiplying the probability of being produced at Plant B (50%) with the probability of being defective at Plant B (4%). This gives us 0.5 * 0.04 = 0.02, which is equal to .02.

Explanation

The probability that an item was produced at Plant B and is defective can be calculated by multiplying the probability of being produced at Plant B (50%) with the probability of being defective at Plant B (4%). This gives us 0.5 * 0.04 = 0.02, which is equal to .02.

8. A study has been done to determine whether or not a certain drug leads to an improvement in symptoms for patients with a particular medical condition. The results are shown in the following table.

	Improvement	No Improvement	Total
Drug	270	530	800
No Drug	120	280	400
Total	390	810	1200

Based on this table, what is the (empirical) probability that a patient shows improvement if it is known that the patient was given the drug?

None of the answers given

.225

.3375

.325

The empirical probability can be calculated by dividing the number of patients who showed improvement after taking the drug by the total number of patients who were given the drug. In this case, the number of patients who showed improvement after taking the drug is 270, and the total number of patients given the drug is 800. Therefore, the empirical probability is 270/800 = 0.3375, which can be expressed as 0.3375 or 33.75%.

Explanation

The empirical probability can be calculated by dividing the number of patients who showed improvement after taking the drug by the total number of patients who were given the drug. In this case, the number of patients who showed improvement after taking the drug is 270, and the total number of patients given the drug is 800. Therefore, the empirical probability is 270/800 = 0.3375, which can be expressed as 0.3375 or 33.75%.

9. A game show requires you to randomly pick 1 of 6 envelopes. The host has hidden a $100 bill in one envelope, and a $1 bill in each of the other 5. Once you've picked, the host is required to remove 2 $1 envelopes you didn't pick. You now have a choice of keeping your original envelope, or paying $2 to switch your choice to one of the 3 you didn't pick.
If you choose to pay $2 and switch your choice, then what are your odds of losing money?

2/3

11/16

13/18

3/4

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Explanation

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10. Suppose E and F are events in a sample space S. Suppose further that E has probability .5, F has probability .6, and the intersection of E and F has probability .2. Find the probability of the union of E and (the complement of F)

.6

.8

.3

.5

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Explanation

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11. Eight cardboard boxes are standing on the table. Two among them contain a present, the other six are empty. You are allowed to open two boxes. How much chance do you have to find at least one present?

7/16

9/16

15/28

13/28

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Explanation

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