Theoretical and experimental probability Problem set A Worked solutions

A fair 10-sided die is numbered 1 to 10. It is rolled once. Find the probability that the score is

1. (a) a factor of 12

   • n(U) = 10
   • Favourable: 1, 2, 3, 4, 6, so n(A) = 5
   • \(P=\frac{1}{2}\)

2. (b) greater than 7

   • n(U) = 10
   • Favourable: 8, 9, 10, so n(A) = 3
   • \(P=\frac{3}{10}\)

The probability that no goals are scored in a hockey match is 0.18. Find the probability that at least one goal is scored.

• At least one is the complement of none.
• \(P(\text{at least one})=1-P(\text{none})=1-0.18\)
• = 0.82

A basketball player scores a free throw with probability 0.72. She takes 150 free throws. Find the expected number of scores.

• Expected number = P × number of trials
• 0.72 × 150 = 108

A random sample of 80 batteries is tested and 6 are faulty. Estimate the probability that a battery is faulty.

• Estimate = number with the property ÷ sample size.
• \(\frac{6}{80}=\frac{3}{40}\) = 0.075

The 11 letters of the word MATHEMATICS are written on separate cards. One card is chosen at random. Find the probability that

1. (a) the letter is M

   • n(U) = 11 letters
   • The letter M appears 2 times.
   • \(P=\frac{2}{11}\)

2. (b) the letter is a vowel

   • n(U) = 11 letters
   • The vowels are A, E, A, I: 4.
   • \(P=\frac{4}{11}\)

In a random sample of 200 workers, 46 cycle to work. A town has 3500 workers.

1. (a) Estimate the probability that a worker cycles to work.

   • \(\frac{46}{200}=\frac{23}{100}\) = 0.23

2. (b) Estimate how many of the town's workers cycle to work.

   • Expected number = 0.23 × 3500 = 805

A fair coin is tossed 4 times. Find the probability of at least one head.

• \(P(\text{none})=\left(\frac{1}{2}\right)^{4}=\frac{1}{16}\)
• \(P(\text{at least one})=1-\frac{1}{16}=\frac{15}{16}\)

A fair six-sided die is rolled 3 times. Find the probability of at least one six.

• \(P(\text{none})=\left(\frac{5}{6}\right)^{3}=\frac{125}{216}\)
• \(P(\text{at least one})=1-\frac{125}{216}=\frac{91}{216}\)

A player wins a game with probability 0.18. The expected number of wins is 27. How many games does the player play?

• Expected number = P × n, so 27 = 0.18 × n
• n = 27 ÷ 0.18 = 150