Conditional Probability Quiz

1.

The letters of the name SMISS are arranged in a line. If an arrangement is chosen at random, what is the probability that the three Ss are together?
- A.
  
  0.6
- B.
  
  0.3
- C.
  
  0.1
- D.
  
  0.4
Correct Answer
B. 0.3

Explanation
The probability that the three Ss are together can be calculated by considering them as a single unit. There are 5 letters in total, and if the three Ss are considered as one unit, there are now 3 units to arrange. The total number of arrangements is 3!, which is 6. Out of these arrangements, only one has the three Ss together. Therefore, the probability is 1/6, which is equal to 0.1667. None of the given answer options match this probability, so the correct answer is not available.

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7
2
2.

A box contains 36 marbles. If a marble is picked at random, the probability of being red is 2/9. How many red marbles should be added to make this probability 1/3?
- A.
  
  6
- B.
  
  4
- C.
  
  12
- D.
  
  9
Correct Answer
A. 6

Explanation
To find the number of red marbles that should be added to make the probability 1/3, we need to determine the total number of marbles after adding the red marbles. We know that the probability of picking a red marble is currently 2/9. To make the probability 1/3, we need to add an equal number of red marbles and non-red marbles. So, if we add 6 red marbles, the total number of marbles will be 42 (36 original marbles + 6 red marbles). The probability of picking a red marble from 42 marbles will be 12/42, which simplifies to 1/3. Therefore, the correct answer is 6.

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1
6
3.

An identity card is picked at random. What is the probability that the sum of the last two digits of its number is 9?
- A.
  
  0.5
- B.
  
  0.4
- C.
  
  0.1
- D.
  
  0.2
Correct Answer
C. 0.1

Explanation
The probability that the sum of the last two digits of the identity card number is 9 can be determined by counting the number of possible combinations that result in a sum of 9 and dividing it by the total number of possible combinations. Since there are 10 possible values for each of the last two digits (0-9), the number of combinations that result in a sum of 9 is 1 (9+0). Therefore, the probability is 1/100, which simplifies to 0.01 or 0.1 in decimal form.

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2
1
4.

One of the 5 points (3, 2), (2, 1), (1, - 4), (5, 5) and (4, 6) is selected at random. What is the probability that it lies on the straight line 3x – 2y = 5?
- A.
  
  0.4
- B.
  
  0.2
- C.
  
  0.1
- D.
  
  0.5
Correct Answer
A. 0.4

Explanation
The equation of the straight line is 3x - 2y = 5. To determine if a point lies on this line, we substitute the x and y coordinates of each point into the equation. We find that only the point (2, 1) satisfies the equation, as 3(2) - 2(1) = 6 - 2 = 4, which is not equal to 5. Therefore, there is only 1 point out of the 5 given points that lies on the line. Since the point is selected at random, the probability of selecting this point is 1/5, which is equal to 0.2. Hence, the correct answer is 0.2.

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1
0
5.

The letters of the word “PROBABILITY” are written on cards and the cards are then shuffled. If a card is picked at random, find the probability that it will contain a vowel.
- A.
  
  5/11
- B.
  
  3/11
- C.
  
  7/11
- D.
  
  4/11
Correct Answer
D. 4/11

Explanation
The word "PROBABILITY" has 11 letters. Out of these 11 letters, there are 4 vowels (O, A, I, and Y). Therefore, the probability of picking a card that contains a vowel is 4/11.

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6.

I have two 10-cent coins, three 20-cent coins, four 50-cent coins and five $1 coins in my pocket. If I choose a coin at random, find the probability that the coin is worth at least 50 cents.
- A.
  
  4/7
- B.
  
  3/7
- C.
  
  9/14
- D.
  
  5/14
Correct Answer
C. 9/14

Explanation
The probability of choosing a coin worth at least 50 cents can be found by dividing the number of coins worth at least 50 cents by the total number of coins. There are four 50-cent coins and five $1 coins, making a total of nine coins worth at least 50 cents. The total number of coins is 2 + 3 + 4 + 5 = 14. Therefore, the probability is 9/14.

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7.

A man tosses two fair dice. One is numbered 1 to 6 in the usual way and the other is numbered 1, 3, 5, 7, 9 and 11. Find the probability that the total of the two numbers shown is greater than 10.
- A.
  
  7/12
- B.
  
  5/12
- C.
  
  1/2
- D.
  
  1/4
Correct Answer
B. 5/12

Explanation
When two dice are tossed, the first die can show any number from 1 to 6, and the second die can show any number from 1, 3, 5, 7, 9, or 11. To find the probability that the total of the two numbers shown is greater than 10, we need to count the number of favorable outcomes and divide it by the total number of possible outcomes. The favorable outcomes are when the first die shows a 5 or 6 and the second die shows a 5 or 7 or 9 or 11. There are 2 favorable outcomes out of 12 possible outcomes, so the probability is 2/12, which simplifies to 1/6. However, this is not one of the answer choices. The closest answer choice is 5/12, which is the complement of 1/6. Therefore, the correct answer is 5/12.

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8.

A card is drawn at random from a normal pack of 52 cards. If A represents the event that the card drawn is a Queen and B represents the event that the card drawn is a Heart. Find P(A|B).
- A.
  
  5/13
- B.
  
  1/4
- C.
  
  4/13
- D.
  
  1/52
Correct Answer
C. 4/13

Explanation
The probability of drawing a Queen given that the card drawn is a Heart can be found using the formula for conditional probability: P(A|B) = P(A∩B) / P(B).
Since there are 4 Queens in a deck of 52 cards, the probability of drawing a Queen is 4/52.
Since there are 13 Hearts in a deck of 52 cards, the probability of drawing a Heart is 13/52.
To find the probability of drawing a Queen given that the card drawn is a Heart, we need to find the probability of drawing a Queen and a Heart, which is 1/52.
Therefore, P(A|B) = (1/52) / (13/52) = 1/13 = 4/52 = 4/13.

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2
9.

Two fair dice, one red and one blue, are tossed. What is the probability that the total of the numbers shown by the two dice exceeds 3?
- A.
  
  1/3
- B.
  
  1/4
- C.
  
  11/12
- D.
  
  1/12
Correct Answer
C. 11/12

Explanation
The probability of the total of the numbers shown by the two dice exceeding 3 can be found by calculating the number of favorable outcomes divided by the total number of possible outcomes. In this case, the favorable outcomes are when the sum of the numbers shown is 4, 5, 6, 7, 8, 9, 10, 11, or 12. There are 9 favorable outcomes out of a total of 36 possible outcomes (since each die has 6 sides). Therefore, the probability is 9/36, which simplifies to 1/4.

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10.

A bag contains 30 balls. The balls are numbered 1,2,3,4 ….30. A ball is drawn at random. Find the probability that the number on the ball is a prime number.
- A.
  
  8/9
- B.
  
  2/3
- C.
  
  1/3
- D.
  
  4/9
Correct Answer
C. 1/3

Explanation
In this question, we are given a bag containing 30 balls numbered from 1 to 30. We need to find the probability that the number on the ball drawn at random is a prime number.

To calculate the probability, we need to determine the number of favorable outcomes (prime numbers) and the total number of possible outcomes (all numbers from 1 to 30).

The prime numbers between 1 and 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. There are a total of 10 prime numbers.

Therefore, the probability of drawing a prime number is 10/30, which simplifies to 1/3.

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1
11.

An interview with 18 people revealed that 5 of the 8 women and 8 of the 10 men preferred drinking coffee to tea. What is the probability that if one person is selected from the group of 18 people, it is either a woman or someone who preferred to drink coffee than tea?
- A.
  
  8/9
- B.
  
  1/3
- C.
  
  1/9
- D.
  
  2/3
Correct Answer
A. 8/9

Explanation
The probability that if one person is selected from the group of 18 people, it is either a woman or someone who preferred to drink coffee than tea can be found by calculating the number of favorable outcomes divided by the total number of possible outcomes. In this case, there are 8 women who preferred drinking coffee, and there are also men who preferred drinking coffee. Since all women are included in the group of 18 people, the total number of favorable outcomes is 8 + 8 = 16. The total number of possible outcomes is 18. Therefore, the probability is 16/18, which simplifies to 8/9.

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1
12.

A computer generates 4-digit random numbers in the range 0000 to 9999 inclusive. Find the probability that the computer produces a random number that begins and ends with the digit 1.
- A.
  
  1/10
- B.
  
  1/4
- C.
  
  1/1000
- D.
  
  1/100
Correct Answer
D. 1/100

Explanation
The probability of the computer producing a random number that begins and ends with the digit 1 can be calculated by considering the number of favorable outcomes (random numbers that begin and end with 1) divided by the total number of possible outcomes (all 4-digit random numbers).

The favorable outcomes can be determined by fixing the first and last digit as 1, and allowing the remaining two digits to vary from 0 to 9. So, there are 10 choices for each of the remaining two digits, resulting in a total of 10 * 10 = 100 favorable outcomes.

The total number of possible outcomes is 10,000, as there are 10 choices for each of the four digits.

Therefore, the probability is 100/10,000 = 1/100.

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Conditional Probability Quiz

The letters of the name SMISS are arranged in a line. If an arrangement is chosen at random, what is the probability that the three Ss are together?

A box contains 36 marbles. If a marble is picked at random, the probability of being red is 2/9. How many red marbles should be added to make this probability 1/3?

An identity card is picked at random. What is the probability that the sum of the last two digits of its number is 9?

One of the 5 points (3, 2), (2, 1), (1, - 4), (5, 5) and (4, 6) is selected at random. What is the probability that it lies on the straight line 3x – 2y = 5?

The letters of the word “PROBABILITY” are written on cards and the cards are then shuffled. If a card is picked at random, find the probability that it will contain a vowel.

I have two 10-cent coins, three 20-cent coins, four 50-cent coins and five $1 coins in my pocket. If I choose a coin at random, find the probability that the coin is worth at least 50 cents.

A man tosses two fair dice. One is numbered 1 to 6 in the usual way and the other is numbered 1, 3, 5, 7, 9 and 11. Find the probability that the total of the two numbers shown is greater than 10.

A card is drawn at random from a normal pack of 52 cards. If A represents the event that the card drawn is a Queen and B represents the event that the card drawn is a Heart. Find P(A|B).

Two fair dice, one red and one blue, are tossed. What is the probability that the total of the numbers shown by the two dice exceeds 3?

A bag contains 30 balls. The balls are numbered 1,2,3,4 ….30. A ball is drawn at random. Find the probability that the number on the ball is a prime number.

An interview with 18 people revealed that 5 of the 8 women and 8 of the 10 men preferred drinking coffee to tea. What is the probability that if one person is selected from the group of 18 people, it is either a woman or someone who preferred to drink coffee than tea?

A computer generates 4-digit random numbers in the range 0000 to 9999 inclusive. Find the probability that the computer produces a random number that begins and ends with the digit 1.