Sample Questions on Probability Rules

Question 1: You flip a fair coin twice. What is the probability of getting exactly one head?

Solution:

To solve this question, we can use the complement rule.

Probability of getting exactly one head is the complement of getting either two heads or two tails.

Probability of getting two heads: [Tex]P(\text{HH}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}[/Tex]

Probability of getting two tails: [Tex]P(\text{TT}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}[/Tex]

So, the probability of getting exactly one head is:

[Tex]1 – P(\text{HH}) – P(\text{TT}) = 1 – \frac{1}{4} – \frac{1}{4} = \frac{1}{2}[/Tex]

Question 2: A bag contains 5 red balls and 3 blue balls. You randomly select two balls from the bag without replacement. What is the probability of selecting one red ball and one blue ball?

Solution:

We can solve this question using the multiplication rule.

First, we find the probability of selecting a red ball, then the probability of selecting a blue ball after selecting a red ball.

[Tex]P(\text{red}) = \frac{5}{8}[/Tex]

[Tex]P(\text{blue after red}) = \frac{3}{7}[/Tex]

So, the probability of selecting one red ball and one blue ball is:

[Tex]P(\text{red and blue}) = P(\text{red}) \times P(\text{blue after red}) = \frac{5}{8} \times \frac{3}{7} = \frac{15}{56}[/Tex]

Question 3: You roll a fair six-sided die. What is the probability of rolling a number divisible by 2 or 3?

Solution:

We can solve this question using the addition rule.

We find the probability of rolling a number divisible by 2, the probability of rolling a number divisible by 3, and subtract the probability of rolling a number divisible by both 2 and 3 (i.e., 6).

Probability of rolling a number divisible by 2: [Tex]P(\text{divisible by 2}) = \frac{3}{6} = \frac{1}{2}[/Tex]

Probability of rolling a number divisible by 3: [Tex]P(\text{divisible by 3}) = \frac{2}{6} = \frac{1}{3}[/Tex]

Probability of rolling a number divisible by both 2 and 3: [Tex]P(\text{divisible by 6}) = \frac{1}{6}[/Tex]

So, the probability of rolling a number divisible by 2 or 3 is:

[Tex]P(\text{divisible by 2 or 3}) = P(\text{divisible by 2}) + P(\text{divisible by 3}) – P(\text{divisible by 6}) = \frac{1}{2} + \frac{1}{3} – \frac{1}{6} = \frac{5}{6}[/Tex]

Question 4: A card is drawn from a standard deck of 52 playing cards. What is the probability of drawing a face card or a heart?

Solution:

We can solve this question using the addition rule.

We find the probability of drawing a face card, the probability of drawing a heart, and subtract the probability of drawing a card that is both a face card and a heart.

Probability of drawing a face card: [Tex]P(\text{face card}) = \frac{12}{52}[/Tex] (There are 3 face cards in each suit)

Probability of drawing a heart: [Tex]P(\text{heart}) = \frac{13}{52}[/Tex]

Probability of drawing a card that is both a face card and a heart: [Tex]P(\text{face card and heart}) = \frac{3}{52}[/Tex] (There are 3 face cards that are hearts)

So, the probability of drawing a face card or a heart is:

[Tex]P(\text{face card or heart}) = P(\text{face card}) + P(\text{heart}) – P(\text{face card and heart}) = \frac{12}{52} + \frac{13}{52} – \frac{3}{52} = \frac{22}{52} = \frac{11}{26}[/Tex]

Question 5: Two dice are rolled. What is the probability of getting a sum greater than 9?

Solution:

We can solve this question by listing all possible outcomes of rolling two dice and counting the outcomes where the sum of the numbers is greater than 9.

There are 6×6 = 36 possible outcomes when rolling two dice. Outcomes with a sum greater than 9:

(4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)

So, there are 6 outcomes where the sum is greater than 9.Therefore, the probability of getting a sum greater than 9 is:

[Tex]P(\text{sum > 9}) = \frac{6}{36} = \frac{1}{6}[/Tex]

Probability is a crucial concept in mathematics and statistics that enables us to quantify uncertainty and make informed decisions in various fields. Whether it’s predicting the outcome of an experiment or assessing risk in finance, probability rules provide a systematic way to analyze and understand random phenomena.

In this article, we’ll explore the fundamental principles of probability, starting with its definition and then delving into the essential rules governing its calculations.