Probability/Stats Worksheet 1 – Basic Probability and Random Variables

  1. [GCSE Revision!] There are two bags A and B. Both bags contain 3 red and 4 blue balls each. I randomly take a ball from bag A and put it in bag B. I then randomly select a ball from bag B. Find the probability that the ball I selected from bag B is blue.
  2. A random variable X represents the number of hours that a student revises for an exam:

/ 0 / 1 / 2 / 3
/ 0.3 / 0.2 / 0.4 / 0.1

Find (a) the expected number of hours revised , and (b) the variance.

  1. Bob wants to model the probability of a given 100m runner getting a particular time using the probability density function . All runners’ times are guaranteed to be between 9s and 18s. Determine the constant .
  2. A random variable represents the width of ants, which can be in the range 0cm to 1cm, with probability density . Determine:
  3. The Canadian Transport Minister determines that the probability of a particular reaction time between spotting a moose on the road and applying the brakes, degraded linearly with time. We can assume no one would have a reaction time greater than 4s. What is the probability that a randomly chosen person’s reaction time is less than a second?
  4. Show that , i.e. that the variance/spread is unaffected by adding 1 to all our values. (It may be helpful to recall that )
  5. [Source: STEP] Recall from the algebra slides that for all real .
  6. Carol has two bags of sweets. The first bag contains red sweets and blue sweets, whereas the second bag contains red sweet and blue sweets, where and are positive integers. Carol shakes the bags and picks one sweet from each bag without looking. Prove that the probability that the sweets are of the same colour cannot exceed the probability that they are of different colours.
  7. Simon has three bags of sweets. The first bag contains red sweets, white sweets and yellow sweets, where , and are positive integers. The second bag contains red sweets, white sweets and yellow sweets. The third bag contains red sweets, white sweets and yellow sweets. Simon shakes the bags and picks one sweet from each bag without looking. Show that the probability that exactly two of the sweets are o the same colour is:
    and find the probability that the sweets are all of the same colour.
  8. Deduce from the scenario above that the probability that exactly two of the sweets are of the same colour is at least 6 times the probability that the sweets are all of the same colour.

Probability/Stats Worksheet 1 – Basic Probability and Random Variables- ANSWERS

  1. By constructing an appropriate tree diagram, or just going straight for the probabilities, we get .

  2. .
  3. (since the area under the PDF must be 1, and times are restricted between 9s and 18s). Integrating, we get , and thus .
  4. cm
    So
  5. We get a PDF looking like this:

    The is to ensure the area under the triangle is 1. At a time of 1s, the probability density is . We need to find the probability mass under the graph when . Finding the area of the resulting trapezium, we get .
    Alternatively, you could have found the equation of the line: . Then we just need to find , which is .
  6. Answers:

  7. Since (and is positive) then .
  8. The probability that two are the same:
  9. If , then and similarly, and . So: