NeokaalStudy

Probability: Competency Practice

Two-dice sum combinations, 3-coin games, deck removals, numbered discs, and geometric probability.

6 problems·20–25 min·★★★★☆
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  1. Problem 1
    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.
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  2. Problem 2
    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.
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  3. Problem 3
    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 →
  4. Problem 4
    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.
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  5. Problem 5
    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.
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  6. Problem 6
    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 →