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Neokaal Study

Probability

Practice Worksheet

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

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Answer all questions.

Show all necessary working.

Use a separate notebook for your solutions.

  1. State the probability of an impossible event, a certain event, and the sum of probabilities of all elementary outcomes.

  2. Which cannot be a probability: $2/3$, $-0.4$, $15\%$, or $0.7$? Explain.

  3. A bag contains only blue counters. Find the probabilities of drawing a red counter and of drawing a blue counter.

  4. A bag contains 7 red and 4 black counters. Find $P(\text{red})$ and $P(\text{not red})$.

  5. A box has 4 red, 6 white, and 2 green marbles. Find the probabilities of red, white, and not green.

  6. 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.

  7. A fair die is rolled once. Find the probabilities of a prime number, a number strictly between 2 and 6, and an odd number.

  8. 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.

  9. 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.

  10. A fair die has faces labelled A, B, C, D, E, A. Find the probabilities of A and D.

  11. 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.

  12. If $P(E)=\frac{8}{10}$, find $P(\overline E)$.

  13. A lot contains 132 good pens and 12 defective pens. Find the probability that a randomly selected pen is good.

  14. 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.

  15. 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.

  16. 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.

  17. 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.

  18. Three fair coins are tossed. A player wins when all three show the same face. Find the probability that the player loses.

  19. A fair die is rolled twice. Find the probabilities that 5 appears neither time and that 5 appears at least once.

  20. Two fair coins are tossed. Are the events 'two heads', 'two tails', and 'one of each' equally likely? Give their probabilities.

Neokaal Study

Answer Key

  1. $0, 1,$ and $1$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  2. $-0.4$, because probabilities lie in $[0,1]$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  3. $0$ and $1$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  4. $\frac{7}{11}$ and $\frac{4}{11}$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  5. $\frac{4}{12}, \frac{6}{12}, \frac{10}{12}$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  6. $1/8, 1/2, 3/4,$ and $1$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  7. $1/2, 1/2,$ and $1/2$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  8. $1/26, 3/13, 3/26,$ and $1/4$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  9. $9/10, 1/10,$ and $1/5$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  10. $1/3$ and $1/6$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  11. $\pi/24$

    1. List or count the equally likely elementary outcomes.
    2. Divide favourable outcomes by total outcomes and simplify.

  12. $\frac{2}{10}$

    1. Complementary probabilities add to 1.
    2. Count one event directly and subtract from 1 when that is shorter.

  13. $11/12$

    1. Complementary probabilities add to 1.
    2. Count one event directly and subtract from 1 when that is shorter.

  14. $31/36$ and $5/36$

    1. Complementary probabilities add to 1.
    2. Count one event directly and subtract from 1 when that is shorter.

  15. $1/5, 1/4,$ and $0$

    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.

  16. $15/19$

    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.

  17. $6/36=1/6$; the sums have different numbers of ordered pairs

    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.

  18. $6/8=3/4$

    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.

  19. $25/36$ and $11/36$

    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.

  20. No; $1/4, 1/4,$ and $1/2$

    1. Write the ordered sample space or count it systematically.
    2. Count favourable outcomes, divide by the total, and use complements where helpful.