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IB Mathematics: Applications and Interpretation HL · 13.1: Probability · 13.1.7

Name Date

Independent events Problem set A

Show every step. Give probabilities as fractions or decimals, and use a sample space diagram or Venn diagram where the question asks.

Difficulty: first skills applying them stretch

Skills

  1. A shop's card reader is out of order on 0.45 of days and its till printer is out of order on 0.6 of days. The two faults are independent. Find the probability that both are out of order on the same day.

  2. For events A and B, \(P(A)=0.5\), \(P(B)=0.3\) and \(P(A\cup B)=0.65\). Determine whether A and B are independent. Justify your answer.

  3. A website's search bar and its login page fail independently. The probability that the search bar fails is 0.4. The probability that both fail is 0.14. Find the probability that the login page fails.

  4. Events A and B are independent, with \(P(A)=\frac{3}{8}\) and \(P(A\cap B)=\frac{3}{20}\). Find \(P(B)\).

  5. Events A and B have P(A) = 0.5 and P(B) = 0.2.

    1. (a) A and B are independent. Find P(A ∪ B).
    2. (b) State, with a reason, whether A and B could also be mutually exclusive.

In context

  1. In a board game, two fair six-sided dice are rolled and the score is the product of the two numbers.

    1. (a) Use a sample space diagram to find the probability that the score is at least 15.
    2. (b) The game is played 144 times. How many times would you expect the score to be at least 15?

  2. A fair four-sided die (numbered 1 to 4) and a fair eight-sided die (numbered 1 to 8) are rolled and the scores are added.

    1. (a) Find the probability that the total is 6.
    2. (b) Find the probability that the total is a multiple of 4.

  3. A sports club has 120 members. Of these, 45 swim, 50 cycle and 15 do both.

    1. (a) Draw a Venn diagram to show this information.
    2. (b) Find the probability that a person chosen at random does neither.
    3. (c) Find the probability that a member swims, given that they do not cycle.

  4. One card is chosen at random from a standard pack of 52 cards. H is the event "the card is a heart" and K is the event "the card is a king".

    1. (a) Are H and K mutually exclusive? Give a reason.
    2. (b) Are H and K independent? Show your working.

Stretch

  1. Two events are independent, with \(P(A)=0.3\) and \(P(B)=0.5\). Find \(P(A\cup B)\).

  2. Events A and B are independent, with \(P(A)=0.6\) and \(P(A\cup B)=0.82\). Find \(P(B)\).

jmaths.xyz · AI HL · 13.1.7 Independent events