Worksheet Chapter 5: Probability Distribution

1. The continuous random variable Z has a standard normal distribution Z ~ N(0, 1). (a) Find P(Z > 1.24). (b) Find P(-0.8 < Z < 1.24).

2. The lifespan of light bulbs produced by a company is normally distributed with a mean of μ hours and a standard deviation of 120 hours. Given that 2.28% of light bulbs have a lifespan exceeding 1240 hours, (a) find the value of μ, (b) hence, find the probability that a light bulb chosen at random has a lifespan of less than 900 hours.

3. The masses of pineapples produced in an orchard follow a normal distribution with a mean of 1.5 kg and a standard deviation of 0.3 kg. (a) A pineapple is chosen at random. Find the probability that its mass is greater than 1.86 kg. (b) If 500 pineapples are harvested, estimate the number of pineapples that weigh between 1.2 kg and 1.86 kg.

4. Fill in the blank: If a binomial random variable X ~ B(n, p) has mean 15 and variance 6, the value of q (probability of failure) is ________.

5. In a driving test, the probability that a candidate passes on the first attempt is 0.7. If a group of 10 candidates take the test independently, calculate the probability that at least 9 candidates pass.

6. Given a continuous random variable X which is normally distributed as X ~ N(50, 16), which expression gives the standardized Z score for X = 58?

7. State whether the following statement is True or False: 'For a normal distribution, the probability of obtaining an exact single value X = k is always equal to 0, i.e., P(X = k) = 0.'

8. The random variable X ~ B(n, p) has a mean of 20 and a variance of 12. (a) Find the values of n and p. (b) Find P(X = 1).

9. It is known that 20% of pens in a box are defective. A sample of 6 pens is chosen at random. Calculate the probability that (a) exactly 2 pens are defective, (b) no pens are defective.

10. A discrete random variable X has probability distribution function P(X = x) = c(5 - x) for x = 1, 2, 3, 4. Find the value of the constant c.