Solving Problems On Provability

 

Understanding Probability

Definition: Probability is the mathematical measure of the likelihood that a specific event will occur. It quantifies uncertainty on a scale from 0 (impossible event) to 1 (certain event), often expressed as a fraction, decimal, or percentage.

Necessary Formulas

1. Theoretical Probability: Used when all outcomes in a sample space are equally likely.

P(A) = (Number of favorable outcomes) / (Total number of possible outcomes)

2. Complement of an Event: The probability of an event not happening.

P(A') = 1 - P(A)

3. Experimental Probability: Based on actual trials or historical data.

P(E) = (Number of times event occurs) / (Total number of trials)

---

Grade 10 Probability Practice Quiz

Test your knowledge with these 10 multiple-choice questions aligned with the Canadian curriculum.

Question 1: A standard Canadian deck of 52 playing cards is well-shuffled. What is the theoretical probability of randomly drawing a Heart?

• A) 1/2
• B) 1/4
• C) 1/13
• D) 1/52

Correct Answer: B) 1/4
Solution: There are 4 suits in a deck (Hearts, Diamonds, Clubs, Spades), each with 13 cards.
P(Heart) = 13 / 52 = 1/4 (or 25%).

Question 2: A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?

• A) 1/3
• B) 1/2
• C) 2/3
• D) 1/6

Correct Answer: A) 1/3
Solution: The possible outcomes are {1, 2, 3, 4, 5, 6} (Total = 6). The numbers greater than 4 are 5 and 6 (Favorable = 2).
P(>4) = 2 / 6 = 1/3.

Question 3: The probability that it will snow in Calgary tomorrow is 0.35. What is the probability that it will NOT snow tomorrow?

• A) 0.35
• B) 0.55
• C) 0.65
• D) 1.35

Correct Answer: C) 0.65
Solution: Using the complement formula: P(Not Snow) = 1 - P(Snow).
P(Not Snow) = 1 - 0.35 = 0.65.

Question 4: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is drawn at random, what is the probability that it is green?

• A) 1/5
• B) 1/2
• C) 1/10
• D) 2/5

Correct Answer: A) 1/5
Solution: Total marbles = 5 + 3 + 2 = 10. Green marbles = 2.
P(Green) = 2 / 10 = 1/5 (or 20%).

Question 5: Two fair coins are tossed at the same time. What is the probability of getting exactly one Head and one Tail?

• A) 1/4
• B) 1/2
• C) 3/4
• D) 1/3

Correct Answer: B) 1/2
Solution: The sample space for two coins is {HH, HT, TH, TT} (Total = 4). The favorable outcomes are HT and TH (Favorable = 2).
P(One H, One T) = 2 / 4 = 1/2.

Question 6: In a standard Toronto high school class, 12 students play soccer, 8 play basketball, and 4 play both sports. If there are 26 students in total, what is the probability that a randomly chosen student plays soccer OR basketball?

• A) 20/26
• B) 16/26
• C) 24/26
• D) 12/26

Correct Answer: B) 16/26
Solution: Use the Principle of Inclusion-Exclusion: P(A or B) = P(A) + P(B) - P(A and B).
Favorable outcomes = 12 + 8 - 4 = 16.
P(Soccer or Basketball) = 16 / 26 (which simplifies to 8/13).

Question 7: A spinner is divided into 8 equal sectors numbered 1 through 8. What is the probability of spinning a prime number?

• A) 1/2
• B) 3/8
• C) 5/8
• D) 1/4

Correct Answer: A) 1/2
Solution: The prime numbers between 1 and 8 are 2, 3, 5, and 7 (Total of 4 numbers).
P(Prime) = 4 / 8 = 1/2.

Question 8: A box contains 4 red pens and 6 blue pens. A pen is drawn at random, its color is recorded, and it is not replaced. A second pen is then drawn. What is the probability that both pens are red?

• A) 16/100
• B) 2/15
• C) 12/90
• D) 4/25

Correct Answer: B) 2/15
Solution: For the first draw, P(Red 1) = 4/10. Since it's not replaced, there are now 3 red pens left out of 9 total pens.
P(Red 2 given Red 1) = 3/9.
P(Both Red) = (4/10) * (3/9) = 12/90 = 2/15.

Question 9: During a hockey practice, a player takes 50 shots on goal and scores 15 times. What is the experimental probability that the player will score on their next shot?

• A) 3/10
• B) 7/10
• C) 15/35
• D) 1/50

Correct Answer: A) 3/10
Solution: Experimental probability is based on past trials.
P(Score) = 15 / 50 = 3/10 (or 30%).

Question 10: Which of the following values cannot represent the mathematical probability of an event?

• A) 0%
• B) 4/3
• C) 0.99
• D) 2/7

Correct Answer: B) 4/3
Solution: Probability values must always fall strictly within the range of 0 to 1 (inclusive). The fraction 4/3 is equal to 1.33, which is greater than 1 and therefore impossible.