AP Statistics 2.7 Independent Events and Unions of Events- Exam Style Questions - MCQs - New Syllabus
▶️ Answer/Explanation
Coin flips are independent events. This means the outcome of one flip does not affect the outcome of any future flip.
Even though the first four flips were heads, the probability that the fifth flip is heads remains:
\( P(\text{Heads})=\frac{1}{2} \)
A common mistake is to think that tails is “due” after several heads in a row. This is known as the gambler’s fallacy. Since the coin is fair, every flip has a 50% chance of landing heads.
✅ Answer: (B)
▶️ Answer/Explanation
Define the events:
\(A=\{\text{height} > 60\}\)
\(B=\{\text{height} > 70\}\)
\(C=\{\text{height} < 65\}\)
Notice that:
\(A \cap C=\{60 < \text{height} < 65\}\)
Any student in \(A \cap C\) must have a height less than 65 inches, while any student in \(B\) must have a height greater than 70 inches. Therefore, no student can belong to both events simultaneously.
\((A \cap C)\cap B=\varnothing\)
Since the intersection is empty, the events are mutually exclusive.
✅ Answer: (E)
▶️ Answer/Explanation
To determine whether the events are independent, compare the overall probability of not reading a book to the conditional probability given that a person is aged 18–44.
Let:
\(P(\text{No Book})=0.22\)
\(P(18\text{–}44)=0.44\)
\(P(\text{No Book} \cap 18\text{–}44)=0.12\)
Compute the conditional probability:
\(P(\text{No Book}\mid 18\text{–}44)=\frac{0.12}{0.44}\approx0.273\)
Since:
\(0.273 \ne 0.22\)
knowing that a person is aged 18–44 changes the probability of not reading a book in the past year. Therefore, the events are not independent.
The probability increases from 0.22 to approximately 0.27, which matches choice (A).
✅ Answer: (A)