Pre-test: Statistics and Probabilities

0.1 Question 1

What is the probability of drawing a King from a standard deck?

  1. 1/13

Explanation: There are 4 Kings in a 52-card deck → 4/52 = 1/13

0.2 Question 2

What is the probability of drawing a card with a value of 10? (Remember: 10, Jack, Queen, and King each count as 10)

  1. 12/52

Explanation: There are 4 each of 10, J, Q, K → 4 × 3 = 12 cards worth 10

1 Question 3

If one card is drawn at random, what is the probability that it is a Heart?

  1. 1/4

Explanation: There are 13 Hearts out of 52 cards → 13/52 = 1/4

2 Question 4

What is the probability of drawing an Ace or a card with value 2?

  1. 8/52

Explanation: 4 Aces + 4 cards with value 2 = 8 cards → 8/52

3 Question 5

What is the probability of drawing a Spade face card? (Face cards: Jack, Queen, King)

  1. 3/52

Explanation: 1 Jack, 1 Queen, 1 King of Spades = 3 cards → 3/52

4 Question 6

Pick one card randomly. What is the probability of getting a card numbered 3 or 7?

  1. 2/13

Explanation: There are 4 cards with number 3 and 4 cards with number 7 → total = 8 cards → 8/52 = 2/13

5 Question 7

What is the probability of drawing a red card with a value greater than 5?

  1. 18/52

Explanation: Red suits = hearts & diamonds (26 cards). Values greater than 5: 6 to 10, J, Q, K (total = 9 values per suit). → 9 × 2 = 18 red cards with value > 5

6 Question 8

What is the probability that a randomly drawn card is not a face card and not an Ace?

    1. 36/5

Explanation:

  • Face cards = 12 (J, Q, K in each suit)

*Aces = 4

→ Not face, not Ace = 52 - 16 = 36/52

6.1 Question 9

If one card is drawn at random, what is the probability that it is a multiple of 4?

(Use only numbered cards: Ace = 1, 2–10; ignore face cards)

  1. 2/13

Explanation: Multiples of 4: 4 and 8 (4 cards each × 2 values = 8 cards) → 8/52 = 2/13

7 Question 10

What is the probability of getting a heart that has a value less than or equal to 4?

  1. 1/13

Explanation: Heart cards with value 1 to 4 (Ace, 2, 3, 4) → 4 specific heart cards → 4/52 = 1/13

8 Normal Distribution

Let \(X\) be a normal distribution with mean \(\mu = 10\) and variance \(\sigma^2 = 36\).

9 Question 11

What is the probability that \(X < 10\)?

  1. 0.5

Explanation: 10 is the mean, and half the data lies below the mean in a normal distribution.

P(X < 10)

P(X < 10)

10 Question 12

What is the probability that \(X > 22\)?

  1. 0.0228

Explanation: Z = (22 - 10)/6 = 2, P(Z > 2) ≈ 0.0228

P(X > 22)

P(X > 22)

11 Question 13

What is the probability that \(4 < X < 16\)?

  1. 0.68

Explanation: This corresponds to Z between -1 and 1.

P(4 < X < 16)

P(4 < X < 16)

12 Question 14

What is the probability that \(X > 28\)?

  1. 0.0013

Explanation: Z = (28 - 10)/6 = 3, P(Z > 3) ≈ 0.0013

P(X > 28)

P(X > 28)

13 Question 15

What is the probability that \(X < 4\)?

  1. 0.1587

Explanation: Z = (4 - 10)/6 = -1 → P(Z < -1) ≈ 0.1587

P(X < 4)

P(X < 4)

14 General Question

14.1 Question 16

A fair six-sided die is rolled once. What is the probability of getting an even number?

  1. 1/2

Explanation: Even numbers are {2, 4, 6} → 3 out of 6 outcomes → 3/6 = 1/2

14.2 Question 17

A bag contains 3 red balls, 5 blue balls, and 2 green balls. One ball is drawn at random. What is the probability that it is not red?

  1. 7/10

Explanation: Total balls = 10. Not red = blue + green = 5 + 2 = 7 → 7/10

14.3 Question 18

You roll a fair die. What is the probability of getting a number greater than 4?

  1. 2/6

Explanation: Numbers > 4 are {5, 6} → 2 outcomes → 2/6 = 1/3

14.4 Question 19

In a box, there are 6 yellow balls, 4 white balls, and 10 black balls. What is the probability of drawing a white or yellow ball?

  1. 10/20

Explanation: Total = 6+4+10 = 20, yellow or white = 6 + 4 = 10 → 10/20 = 1/2

14.5 Question 20

A die is rolled and a ball is drawn from a bag containing 2 red, 3 blue, and 5 green balls. What is the probability of getting an even number on the die and a green ball?

  1. 1/4

Explanation:

  • P(even number) = 3/6 = 1/2

  • P(green ball) = 5/10 = 1/2

  • Combined = 1/2 × 1/2 = 1/4