Q: A fair coin is tossed 10 times. What is the probability of getting exactly 6 heads? Use the binomial probability formula: $$P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$, where n = 10$, k = 6$, and p = 0.5$.

$$P(X = 6) = \binom{10}{6} (0.5)^6$ (0.5)^4$ = 210 \times (0.5$$)^{10}$$ = 0.205$$

Q: Suppose 30 students take Math, 35 take English, and 15 take both. If there are 50 students in total, what is the probability that a randomly selected student takes Math or English?

$$P(\text{Math or English}) = \frac{30 + 35 - 15}{50} = \frac{50}{50} = 1$$

Q: A fair coin is tossed 4 times. What is the probability of getting all tails?

$$P(\text{all tails}) = (0.5)^4$ = 0.0625$$

Q: In a psychology experiment, the probability that a participant guesses the correct answer by chance is 0.25. If a participant guesses on 3 independent questions, what is the probability they get all 3 correct?

$$P(\text{all correct}) = (0.25)^3$ = 0.015625$$

Question 1

A fair coin is tossed 10 times. What is the probability of getting exactly 6 heads? Use the binomial probability formula: $$P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$, where n = 10$, k = 6$, and p = 0.5$.

Accepted Answer

$$P(X = 6) = \binom{10}{6} (0.5)^6$ (0.5)^4$ = 210 	imes (0.5$$)^{10}$$ = 0.205$$

Question 2

Which of the following statements about mutually exclusive (disjoint) events is TRUE?

Accepted Answer

If events A and B are mutually exclusive, then $$P(A \cap B) = 0$$

Question 3

Suppose 30 students take Math, 35 take English, and 15 take both. If there are 50 students in total, what is the probability that a randomly selected student takes Math or English?

Accepted Answer

$$P(	ext{Math or English}) = \frac{30 + 35 - 15}{50} = \frac{50}{50} = 1$$

Question 4

If the probability of winning a lottery is $$P = 0$$, what does this imply about the event?

Accepted Answer

The event is impossible and will never occur.

Question 5

A Venn diagram shows two events, A and B, with an intersection. Which formula correctly calculates $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Question 6

If two events A and B are independent, which of the following is TRUE?

Accepted Answer

$$P(A \cap B) = P(A) \cdot P(B)$$

Question 7

A hospital reports that the probability a patient has disease A is 0.2, and the probability a patient has disease B is 0.3. If the probability a patient has both diseases is 0.1, what is the probability a patient has either disease A or disease B?

Accepted Answer

$$P(A \cup B) = 0.2 + 0.3 - 0.1 = 0.4$$

Question 8

If $$P(A) = 0.6$$ and $$P(A^c) is $$the probability of the complement of A, what is $$P(A^c)$$?

Accepted Answer

$$P(A^c) = 1 - 0.6 = 0.4$$

Question 9

A fair coin is tossed 4 times. What is the probability of getting all tails?

Accepted Answer

$$P(	ext{all tails}) = (0.5)^4$ = 0.0625$$

Question 10

In a psychology experiment, the probability that a participant guesses the correct answer by chance is 0.25. If a participant guesses on 3 independent questions, what is the probability they get all 3 correct?

Accepted Answer

$$P(	ext{all correct}) = (0.25)^3$ = 0.015625$$

Question 11

Which of the following best describes the Law of Large Numbers?

Accepted Answer

As the number of trials increases, the sample proportion approaches the theoretical probability.

Question 12

If the probability of event A is 0.7 and the probability of event B is 0.5, and A and B are independent, what is the probability that both A and B occur?

Accepted Answer

$$P(A \cap B) = 0.7 	imes 0.5 = 0.35$$

Question 13

A contingency table shows that 40 people like coffee, 30 like tea, and 10 like both. What is the probability that a randomly selected person likes neither coffee nor tea, if there are 70 people in total?

Accepted Answer

$$P(	ext{neither}) = 1 - \frac{40 + 30 - 10}{70} = 1 - \frac{60}{70} = 0.143$$

Question 14

If $$P(B|A) = \frac{P(A \cap B)}{P(A)}$$, what does $$P(B|A)$$ represent?

Accepted Answer

The probability of event B occurring given that event A has occurred.

Question 15

A student is randomly selected from a group where 60% are female and 40% are male. If the probability that a student is left-handed is 10%, and gender and handedness are independent, what is the probability that a randomly selected student is a left-handed female?

Accepted Answer

$$P(	ext{left-handed female}) = 0.6 	imes 0.1 = 0.06$$

Question 16

Which of the following is an example of an impossible event?

Accepted Answer

Being born on February 31st.

Question 17

If two events are disjoint, which of the following must be true?

Accepted Answer

They cannot both occur at the same time.

Question 18

A bag contains 5 red balls and 7 blue balls. If one ball is drawn at random, what is the probability it is red?

Accepted Answer

$$P(	ext{red}) = \frac{5}{12}$$

Question 19

In a coin toss experiment, which method is used to estimate the probability of an event by repeating the experiment many times and recording the frequency of the event?

Accepted Answer

Empirical (experimental) probability

Chapter 5 - Probability Rules and Applications: Coin Tosses, Events, and Conditional Probability

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