Let's solve each of these probability problems step by step and explain the reasoning behind each solution.
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Problem 1:
Six balls numbered from 1 to 6 are placed in an urn. If one ball is selected at random, find the probability that it is an odd-numbered ball.
- Total balls = 6 (numbered 1 through 6)
- Odd numbers: 1, 3, 5 → 3 odd-numbered balls
- Probability = Number of favorable outcomes / Total outcomes
$$
P(\text{odd}) = \frac{3}{6} = \frac{1}{2}
$$
✔ Answer: [B] $\frac{1}{2}$
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Problem 2:
A single six-sided fair die is tossed. Find the probability of obtaining a number greater than 4.
- Numbers on a die: 1, 2, 3, 4, 5, 6
- Numbers > 4: 5, 6 → 2 outcomes
- Total possible outcomes = 6
- Probability:
$$
P(\text{>4}) = \frac{2}{6} = \frac{1}{3}
$$
✔ Answer: [A] $\frac{1}{3}$
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Problem 3:
You are one of 30 people entering a contest. What is the probability that your name will be drawn first?
- Each person has an equal chance to be drawn first.
- Since there are 30 people, your chance is:
$$
P(\text{you first}) = \frac{1}{30}
$$
✔ Answer: [C] $\frac{1}{30}$
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Problem 4:
Given the set of numbers {0, 1, 2, 3, 4, 5, 6, 7, 8}, if one of the numbers is chosen at random, find the probability that the number is a solution of $3x + 1 < 13$.
First, solve the inequality:
$$
3x + 1 < 13 \\
3x < 12 \\
x < 4
$$
So we want values in the set less than 4:
→ {0, 1, 2, 3} → 4 values
Total numbers in set = 9
Probability:
$$
P(x < 4) = \frac{4}{9}
$$
✔ Answer: [B] $\frac{4}{9}$
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Problem 5:
What is the probability of drawing a spade from a deck of 52 playing cards?
- A standard deck has 52 cards.
- There are 13 spades (one for each rank: Ace to King).
- So,
$$
P(\text{spade}) = \frac{13}{52} = \frac{1}{4}
$$
✔ Answer: [D] $\frac{1}{4}$
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Problem 6:
This is a spinner used in a board game Helen invented. The spinner has sections labeled: 10, 12, 18, 22, 24, 36, 54, 64. What is the probability that the spinner will land on a multiple of 3 and 4?
We need to find numbers that are
multiples of both 3 and 4, i.e.,
multiples of LCM(3,4) = 12.
Check which numbers on the spinner are divisible by 12:
- 10 → no
- 12 → yes (12 ÷ 12 = 1)
- 18 → no (18 ÷ 12 = 1.5)
- 22 → no
- 24 → yes (24 ÷ 12 = 2)
- 36 → yes (36 ÷ 12 = 3)
- 54 → no (54 ÷ 12 = 4.5)
- 64 → no
So, multiples of 12:
12, 24, 36 → 3 numbers
Total sections = 8
Probability:
$$
P(\text{multiple of 3 and 4}) = \frac{3}{8}
$$
✔ Answer: [C] $\frac{3}{8}$
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Problem 7:
Joseph has 2 pairs of white socks, 4 pairs of black socks, and 1 pair of blue socks. Socks are not paired together. What is the probability that Joseph will pick a black sock?
- 2 pairs of white = 4 white socks
- 4 pairs of black = 8 black socks
- 1 pair of blue = 2 blue socks
Total socks = 4 + 8 + 2 = 14
Black socks = 8
Probability of picking a black sock:
$$
P(\text{black}) = \frac{8}{14} = \frac{4}{7}
$$
✔ Answer: [C] $\frac{4}{7}$
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✔ Final Answers Summary:
| Problem | Answer |
|--------|--------|
| 1 | [B] $\frac{1}{2}$ |
| 2 | [A] $\frac{1}{3}$ |
| 3 | [C] $\frac{1}{30}$ |
| 4 | [B] $\frac{4}{9}$ |
| 5 | [D] $\frac{1}{4}$ |
| 6 | [C] $\frac{3}{8}$ |
| 7 | [C] $\frac{4}{7}$ |
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Parent Tip: Review the logic above to help your child master the concept of math worksheet for 7th grade with answers.
