Basic Definitions — Hard Practice Quiz

A Probability cheat sheet for Basic Definitions — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Probability of Event E

$$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$$

Probability Range

$$0 \leq P(E) \leq 1$$

Certain and Impossible Events

$$P(\text{certain event}) = 1, \quad P(\text{impossible event}) = 0$$

Complement Rule

$$P(E^c) = 1 - P(E)$$

Where: \(E^c\) = complement of E

Practice quiz

  1. A bag contains red and blue marbles. The probability of drawing a red marble is $P(R)$. If the probability of drawing a blue marble is $P(B)$, and $P(B) = 2P(R)$, what is $P(R)$?

    • $\frac{1}{2}$
    • $\frac{1}{3}$
    • $\frac{2}{3}$
    • $\frac{1}{4}$

    Answer: $\frac{1}{3}$

  2. In a group of people, the probability that a randomly chosen person likes coffee is $0.6$ and the probability that they like tea is $0.5$. If the probability that a person likes neither coffee nor tea is $0.2$, what is the probability that a randomly chosen person likes both coffee and tea?

    • $0.3$
    • $0.4$
    • $0.7$
    • $0.8$

    Answer: $0.3$

  3. An event $A$ has a probability $P(A)$. The probability of its complement, $P(A^c)$, is three times $P(A)$. If the total number of outcomes is $100$, how many favorable outcomes are there for event $A$?

    • $20$
    • $25$
    • $33$
    • $75$

    Answer: $25$

  4. A fair six-sided die is rolled. Let $E$ be the event of rolling an even number. Let $F$ be the event of rolling a number greater than $4$. Which of the following statements is true regarding $P(E^c)$ and $P(F)$?

    • $P(E^c) < P(F)$
    • $P(E^c) = P(F)$
    • $P(E^c) > P(F)$
    • It is impossible to compare them without more information.

    Answer: $P(E^c) > P(F)$

  5. In a lottery, the probability of winning a small prize is $P(S)$, and the probability of winning a large prize is $P(L)$. The probability of winning no prize is $0.7$. If $P(L) = 0.5 \times P(S)$, what is $P(S)$? Assume winning a small prize and winning a large prize are mutually exclusive events.

    • $0.1$
    • $0.2$
    • $0.3$
    • $0.4$

    Answer: $0.2$

  6. If $P(A)$ is the probability of event $A$, and $P(A^c)$ is the probability of its complement, which of the following statements is always true?

    • $P(A) + P(A^c) = 1$
    • $P(A) \times P(A^c) \leq 0.25$
    • If $P(A) > 0.5$, then $P(A^c) < 0.5$
    • All of the above.

    Answer: All of the above.

  7. A box contains $10$ red balls and $N$ blue balls. The probability of drawing a red ball is $P(R)$. If $5$ red balls are added to the box, the new probability of drawing a red ball, $P(R')$, is $1.2$ times $P(R)$. What is the initial number of blue balls, $N$?

    • $5$
    • $10$
    • $15$
    • $20$

    Answer: $10$

  8. If an event $A$ is an impossible event, what can be said about $P(A^c)$?

    • $P(A^c) = 0$
    • $P(A^c) = 0.5$
    • $P(A^c) = 1$
    • It cannot be determined without more information.

    Answer: $P(A^c) = 1$

  9. A survey shows that $P(\text{student owns a laptop}) = 0.8$ and $P(\text{student owns a smartphone}) = 0.9$. The probability that a student owns neither a laptop nor a smartphone is $0.05$. What is the probability that a student owns both a laptop and a smartphone?

    • $0.65$
    • $0.75$
    • $0.85$
    • $0.95$

    Answer: $0.75$

  10. A jar contains only red and green candies. The probability of picking a red candy is $P(R)$. If the number of green candies is $G$ and the total number of candies is $T$, and $P(R^c) = \frac{2}{3}$, what is the ratio of red candies to green candies?

    • $\frac{1}{3}$
    • $\frac{1}{2}$
    • $\frac{2}{3}$
    • $2$

    Answer: $\frac{1}{2}$

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