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IB Mathematics: Applications and Interpretation HL · 13.1: Probability · 13.1.3

Name Date

Tree diagrams and sample spaces Problem set A

Draw a tree diagram, a Venn diagram or a sample space diagram. Multiply along branches and add the end outcomes you want. Give a probability as a fraction in lowest terms, or as a decimal when the numbers are decimals.

Difficulty: first skills applying them stretch

Skills

  1. Every day, a student either walks to school (probability 0.6) or goes by bus. If the student walks, the probability of being late is 0.1. If the student goes by bus, the probability of being late is 0.25. Find the probability that the student

    1. (a) walks and is late.
    2. (b) is late.

  2. A bag contains 4 yellow counters and 6 green counters. Two counters are taken at random, one after the other, without replacement. Find the probability that

    1. (a) both are yellow.
    2. (b) the two are different colours.

  3. The Venn diagram shows probabilities for the students at a school. P is the event "has a phone" and T is the event "has a tablet". A student is chosen at random.

    0.320.270.18xPT
    1. (a) Find the value of x.
    2. (b) Find the probability that the student has a phone.
    3. (c) Find the probability that the student has a tablet but no phone.

  4. Two fair six-sided dice are rolled and the scores are added. Draw a sample space diagram and find the probability that

    1. (a) the sum is 8
    2. (b) the sum is less than 5

  5. Two fair coins are tossed. Find the probability of

    1. (a) exactly one head.
    2. (b) at least one tail.

  6. Three cards show the letters J, A and M. They are placed in a row in a random order. List all the arrangements and find the probability that the first letter is a consonant.

  7. A bag holds 3 red counters and 5 blue counters. A counter is taken at random, its colour is noted and it is put back. A second counter is then taken. Find the probability that

    3/8R3/8R9/645/8B15/645/8B3/8R15/645/8B25/64
    1. (a) both counters are red.
    2. (b) the two counters are different colours.

In context

  1. Machine X makes 60% of the bolts in a factory and machine Y makes the rest. 3% of the bolts from X are defective and 5% of the bolts from Y are defective. A bolt is chosen at random. Find the probability that it

    1. (a) was made by X and is defective.
    2. (b) is defective.

  2. A box holds 5 milk, 4 dark and 3 white chocolates. Two chocolates are taken at random without replacement. Find the probability that both are the same type.

  3. An archer hits the target with each arrow with probability 0.8, independently. She shoots three arrows. Find the probability that she hits the target

    1. (a) exactly twice.
    2. (b) at least once.

Stretch

  1. A drawer holds 10 light bulbs, 3 of which are faulty. Two bulbs are taken without replacement. Find the probability that at least one is faulty.

  2. A fair four-sided die and a fair six-sided die are rolled and the scores are added. Find the probability that the sum is a prime number.

jmaths.xyz · AI HL · 13.1.3 Tree diagrams and sample spaces