Tree diagrams and sample spaces Problem set B Worked solutions

Tomorrow it will rain with probability 0.35. If it rains, the probability that Omar takes the bus is 0.8. If it does not rain, the probability that he takes the bus is 0.3. Find the probability that

1. (a) it rains and Omar takes the bus.

   • Draw the tree: first stage rain or no rain.
   • 0.35 × 0.8 = 0.28

2. (b) Omar takes the bus.

   • The other path: 0.65 × 0.3 = 0.195
   • Add the two paths: 0.28 + 0.195 = 0.475

A box contains 5 black cubes and 4 white cubes. Two cubes are taken at random, one after the other, without replacement. Find the probability that

1. (a) both are black.

   • \(P(\text{black}\text{ then }\text{black})=\frac{5}{9}\times\frac{4}{8}=\frac{5}{18}\)

2. (b) the two are different colours.

   • \(P(\text{black then white})=\frac{5}{9}\times\frac{4}{8}=\frac{5}{18}\)
   • The other order gives the same probability.
   • \(P=2\times\frac{5}{18}=\frac{5}{9}\)

The Venn diagram shows probabilities for the people at a leisure centre. C is the event "uses the café" and B is the event "books a class". A person is chosen at random.

1. (a) Find the value of x.

   • The probabilities add to 1: 0.41 + 0.12 + 0.29 + x = 1
   • x = 0.18

2. (b) Find the probability that the person uses the café.

   • Everything inside the first circle: 0.41 + 0.12 = 0.53

3. (c) Find the probability that the person books a class but does not use the café.

   • Only the second circle: 0.29

Two fair five-sided spinners, each numbered 1 to 5, are spun and the scores are added. Draw a sample space diagram and find the probability that

1. (a) the sum is 7

   • Draw the 5 by 5 grid of sums: 25 equally likely cells.
   • Cells with sum 7: 4
   • \(P=\frac{4}{25}\)

2. (b) the sum is less than 4

   • Cells with sum less than 4: 3
   • \(P=\frac{3}{25}\)

A family has two children. Each child is equally likely to be a boy (B) or a girl (G). Find the probability that the family has

1. (a) exactly one girl.

   • All outcomes: BB, BG, GB, GG
   • Favourable outcomes: 2
   • \(P=\frac{2}{4}=\frac{1}{2}\)

2. (b) at least one boy.

   • Outcomes with at least one B: 3
   • \(P=\frac{3}{4}\)

Three tiles show the letters K, I and T. They are placed in a row in a random order. List all the arrangements and find the probability that the last letter is a vowel.

• All arrangements: KIT, KTI, IKT, ITK, TKI, TIK
• Favourable: KTI, TKI (2)
• \(P=\frac{2}{6}=\frac{1}{3}\)

A jar holds 2 mint sweets and 7 toffee sweets. A sweet is taken at random, its type is noted and it is put back. A second sweet is then taken. Find the probability that

1. (a) both sweets are mint.

   • With replacement the jar is the same for both draws: P(mint) = 2/9 each time.
   • \(\frac29\times\frac29=\frac{4}{81}=0.0494\)

2. (b) the two sweets are different types.

   • MT: \(\frac29\times\frac79=\frac{14}{81}\). TM: \(\frac79\times\frac29=\frac{14}{81}\)
   • Add: \(\frac{28}{81}=0.346\)

A bakery makes 65% white loaves and the rest brown. 4% of the white loaves and 9% of the brown loaves are returned by customers. A loaf is chosen at random. Find the probability that it

1. (a) is white and is returned.

   • 0.65 × 0.04 = 0.026

2. (b) is returned.

   • Other path: 0.35 × 0.09 = 0.0315
   • Add: 0.026 + 0.0315 = 0.0575

A tin holds 4 blue, 3 black and 2 red pens. Two pens are taken at random without replacement. Find the probability that both are the same colour.

• Same type twice: three paths. After one is taken, 8 remain.
• \(P=\frac{4}{9}\times\frac{3}{8} + \frac{3}{9}\times\frac{2}{8} + \frac{2}{9}\times\frac{1}{8}\)
• \(P=\frac{5}{18}\)

A seed company says that each seed it sells grows with probability 0.6, independently. A gardener plants three seeds. Find the probability that

1. (a) exactly two seeds grow.

   • Paths with two growing and one not: GGN, GNG, NGG.
   • Each path: \(0.6\times0.6\times0.4=0.144\)
   • Add: \(3\times0.144=0.432\)

2. (b) at least one seed grows.

   • P(none grows) = \(0.4^3=0.064\)
   • \(1-0.064=0.936\)

A tray holds 8 eggs, 3 of which are cracked. Two eggs are taken without replacement. Find the probability that at least one is cracked.

• \(P(\text{none})=\frac{5}{8}\times\frac{4}{7}=\frac{5}{14}\)
• \(P(\text{at least one})=1-\frac{5}{14}=\frac{9}{14}\)

A fair four-sided die and a fair five-sided spinner are rolled and the scores are added. Find the probability that the sum is a multiple of 3.

• Draw the 4 by 5 grid of sums: 20 cells.
• Cells where the sum is a multiple of 3: 7
• \(P=\frac{7}{20}\)