Theoretical and experimental probability Problem set B Worked solutions

A card is taken at random from 15 cards numbered 1 to 15. Find the probability that the number is

1. (a) a multiple of 4

   • n(U) = 15
   • Favourable: 4, 8, 12, so n(A) = 3
   • \(P=\frac{1}{5}\)

2. (b) greater than 11

   • n(U) = 15
   • Favourable: 12, 13, 14, 15, so n(A) = 4
   • \(P=\frac{4}{15}\)

The probability that a plant produces no flowers in a season is 0.27. Find the probability that it produces at least one flower.

• At least one is the complement of none.
• \(P(\text{at least one})=1-P(\text{none})=1-0.27\)
• = 0.73

A seed germinates with probability 0.85. A gardener plants 240 seeds. Find the expected number that germinate.

• Expected number = P × number of trials
• 0.85 × 240 = 204

A random sample of 150 eggs is checked and 12 are cracked. Estimate the probability that an egg is cracked.

• Estimate = number with the property ÷ sample size.
• \(\frac{12}{150}=\frac{2}{25}\) = 0.08

The 10 letters of the word STATISTICS are written on separate tiles. One tile is chosen at random. Find the probability that

1. (a) the letter is I

   • n(U) = 10 letters
   • The letter I appears 2 times.
   • \(P=\frac{1}{5}\)

2. (b) the letter is a vowel

   • n(U) = 10 letters
   • The vowels are A, I, I: 3.
   • \(P=\frac{3}{10}\)

In a random sample of 250 households, 65 recycle glass. A city has 4800 households.

1. (a) Estimate the probability that a household recycles glass.

   • \(\frac{65}{250}=\frac{13}{50}\) = 0.26

2. (b) Estimate how many of the city's households recycle glass.

   • Expected number = 0.26 × 4800 = 1248

A family has 5 children. Each child is equally likely to be a boy or a girl. Find the probability that at least one is a girl.

• \(P(\text{none})=\left(\frac{1}{2}\right)^{5}=\frac{1}{32}\)
• \(P(\text{at least one})=1-\frac{1}{32}=\frac{31}{32}\)

A fair four-sided spinner is numbered 1 to 4. It is spun 3 times. Find the probability that it lands on 4 at least once.

• \(P(\text{none})=\left(\frac{3}{4}\right)^{3}=\frac{27}{64}\)
• \(P(\text{at least one})=1-\frac{27}{64}=\frac{37}{64}\)

A club member wins a raffle prize with probability 0.35. The expected number of prizes is 56. How many raffles does the member enter?

• Expected number = P × n, so 56 = 0.35 × n
• n = 56 ÷ 0.35 = 160