NeokaalStudy

Probability: Practice Worksheet

Elementary outcomes, complementary events, cards, dice, coins, and compound experiments.

20 problems·35–45 min·★★★★☆
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  1. Problem 1
    theoretical-probability·★★☆☆☆
    State the probability of an impossible event, a certain event, and the sum of probabilities of all elementary outcomes.
    ▶Answer
    $0, 1,$ and $1$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  2. Problem 2
    theoretical-probability·★★★☆☆
    Which cannot be a probability: $2/3$, $-0.4$, $15\%$, or $0.7$? Explain.
    ▶Answer
    $-0.4$, because probabilities lie in $[0,1]$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  3. Problem 3
    theoretical-probability·★★★★☆
    A bag contains only blue counters. Find the probabilities of drawing a red counter and of drawing a blue counter.
    ▶Answer
    $0$ and $1$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  4. Problem 4
    theoretical-probability·★★★★★
    A bag contains 7 red and 4 black counters. Find $P(\text{red})$ and $P(\text{not red})$.
    ▶Answer
    $\frac{7}{11}$ and $\frac{4}{11}$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  5. Problem 5
    theoretical-probability·★★☆☆☆
    A box has 4 red, 6 white, and 2 green marbles. Find the probabilities of red, white, and not green.
    ▶Answer
    $\frac{4}{12}, \frac{6}{12}, \frac{10}{12}$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  6. Problem 6
    theoretical-probability·★★★☆☆
    A fair spinner is numbered 1 through 8. Find the probabilities of 8, an odd number, a number greater than 2, and a number less than 9.
    ▶Answer
    $1/8, 1/2, 3/4,$ and $1$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  7. Problem 7
    theoretical-probability·★★★★☆
    A fair die is rolled once. Find the probabilities of a prime number, a number strictly between 2 and 6, and an odd number.
    ▶Answer
    $1/2, 1/2,$ and $1/2$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  8. Problem 8
    theoretical-probability·★★★★★
    One card is drawn from a standard 52-card deck. Find the probabilities of a red king, any face card, a red face card, and a spade.
    ▶Answer
    $1/26, 3/13, 3/26,$ and $1/4$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  9. Problem 9
    theoretical-probability·★★☆☆☆
    A box contains discs numbered 1 to 90. Find the probabilities of drawing a two-digit number, a perfect square, and a multiple of 5.
    ▶Answer
    $9/10, 1/10,$ and $1/5$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  10. Problem 10
    theoretical-probability·★★★☆☆
    A fair die has faces labelled A, B, C, D, E, A. Find the probabilities of A and D.
    ▶Answer
    $1/3$ and $1/6$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  11. Problem 11
    theoretical-probability·★★★★☆
    A point is chosen uniformly in a $3\text{ m}\times2\text{ m}$ rectangle. Find the probability that it lies inside a circle of diameter 1 m wholly inside the rectangle.
    ▶Answer
    $\pi/24$
    ▶Step-by-step solution
    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.
    Report an issue →
  12. Problem 12
    complementary-events·★★★★★
    If $P(E)=\frac{8}{10}$, find $P(\overline E)$.
    ▶Answer
    $\frac{2}{10}$
    ▶Step-by-step solution
    1. Complementary probabilities add to 1.
    2. Count one event directly and subtract from 1 when that is shorter.
    Report an issue →
  13. Problem 13
    complementary-events·★★☆☆☆
    A lot contains 132 good pens and 12 defective pens. Find the probability that a randomly selected pen is good.
    ▶Answer
    $11/12$
    ▶Step-by-step solution
    1. Complementary probabilities add to 1.
    2. Count one event directly and subtract from 1 when that is shorter.
    Report an issue →
  14. Problem 14
    complementary-events·★★★☆☆
    Of 144 pens, 20 are defective. A buyer accepts a randomly offered pen exactly when it is good. Find the probabilities of acceptance and rejection.
    ▶Answer
    $31/36$ and $5/36$
    ▶Step-by-step solution
    1. Complementary probabilities add to 1.
    2. Count one event directly and subtract from 1 when that is shorter.
    Report an issue →
  15. Problem 15
    compound-experiment·★★★★☆
    Five cards labelled 10, J, Q, K, A are shuffled. Find $P(Q)$ on the first draw. If Q is removed, find $P(A)$ and $P(Q)$ on the second draw.
    ▶Answer
    $1/5, 1/4,$ and $0$
    ▶Step-by-step solution
    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.
    Report an issue →
  16. Problem 16
    compound-experiment·★★★★★
    A lot has 16 good and 4 defective bulbs. A good bulb is drawn and not replaced. Find the probability that the next bulb is good.
    ▶Answer
    $15/19$
    ▶Step-by-step solution
    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.
    Report an issue →
  17. Problem 17
    compound-experiment·★★☆☆☆
    Two distinguishable fair dice are rolled. Find the probability that their sum is 7, and explain why the eleven possible sums are not equally likely.
    ▶Answer
    $6/36=1/6$; the sums have different numbers of ordered pairs
    ▶Step-by-step solution
    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.
    Report an issue →
  18. Problem 18
    compound-experiment·★★★☆☆
    Three fair coins are tossed. A player wins when all three show the same face. Find the probability that the player loses.
    ▶Answer
    $6/8=3/4$
    ▶Step-by-step solution
    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.
    Report an issue →
  19. Problem 19
    compound-experiment·★★★★☆
    A fair die is rolled twice. Find the probabilities that 5 appears neither time and that 5 appears at least once.
    ▶Answer
    $25/36$ and $11/36$
    ▶Step-by-step solution
    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.
    Report an issue →
  20. Problem 20
    compound-experiment·★★★★★
    Two fair coins are tossed. Are the events 'two heads', 'two tails', and 'one of each' equally likely? Give their probabilities.
    ▶Answer
    No; $1/4, 1/4,$ and $1/2$
    ▶Step-by-step solution
    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.
    Report an issue →