Combined and mutually exclusive events Problem set B Worked solutions

Find P(A ∩ B) when:

1. (a) P(A) = 0.65, P(B) = 0.5, P(A ∪ B) = 0.85.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(0.85=0.65+0.5-P(A\cap B)\)
   • \(P(A\cap B)=0.65+0.5-0.85=0.3\)

2. (b) P(A) = 0.4, P(B) = 0.25, P(A ∪ B) = 0.55.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(0.55=0.4+0.25-P(A\cap B)\)
   • \(P(A\cap B)=0.4+0.25-0.55=0.1\)

Find P(A ∪ B) when:

1. (a) P(A) = 0.55, P(B) = 0.3, P(A ∩ B) = 0.2.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(P(A\cup B)=0.55+0.3-0.2=0.65\)

2. (b) P(A) = 0.65, P(B) = 0.4, P(A ∩ B) = 0.25.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(P(A\cup B)=0.65+0.4-0.25=0.8\)

Find P(B) when:

1. (a) P(A) = 0.6, P(A ∩ B) = 0.25, P(A ∪ B) = 0.9.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(0.9=0.6+P(B)-0.25\)
   • \(P(B)=0.9-0.6+0.25=0.55\)

2. (b) P(A) = 0.45, P(A ∩ B) = 0.15, P(A ∪ B) = 0.7.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(0.7=0.45+P(B)-0.15\)
   • \(P(B)=0.7-0.45+0.15=0.4\)

Events A and B are mutually exclusive.

1. (a) P(A) = 0.34 and P(A ∪ B) = 0.8. Find P(B).

   • Mutually exclusive: \(P(A\cap B)=0\), so \(P(A\cup B)=P(A)+P(B)\)
   • \(0.8=0.34+P(B)\)
   • \(P(B)=0.8-0.34=0.46\)

2. (b) P(A) = 1/4 and P(B) = 2/9. Find P(A ∪ B).

   • \(P(A\cup B)=P(A)+P(B)\)
   • \(P(A\cup B)=\frac{1}{4}+\frac{2}{9}\)
   • \(=\frac{17}{36}\)

A and B are two events. P(A) = 0.3, P(B) = 0.45 and P(A ∪ B) = 0.75. Are A and B mutually exclusive? Give a reason.

• \(P(A\cap B)=0.3+0.45-0.75=0\)
• It is 0, so A and B cannot happen together: they are mutually exclusive.

In a city, 38% of flats have a balcony, 22% have a garage and 7% have both. A flat is chosen at random.

1. (a) Find the probability that the flat has a balcony or a garage (or both).

   • \(P(\text{either})=0.38+0.22-0.07=0.53\)

2. (b) Find the probability that the flat has neither.

   • Neither is everything outside the union: \(1-0.53=0.47\)

In a gym, a member is chosen at random. The probability that the member uses the pool is 0.52, and the probability that the member uses the climbing wall is 0.31. The probability that the member uses the pool or the wall (or both) is 0.67.

1. (a) Find the probability that the member uses both.

   • \(0.67=0.52+0.31-P(\text{both})\)
   • \(P(\text{both})=0.52+0.31-0.67=0.16\)

2. (b) Find the probability that the member uses the pool but not the wall.

   • Pool only = 0.52 − 0.16 = 0.36

A pupil walks to school or comes by bus, never both on the same day. The probability of walking is 0.29 and the probability of the bus is 0.46. The two are mutually exclusive.

1. (a) Find the probability that the pupil walks or comes by bus.

   • Mutually exclusive, so add the probabilities.
   • \(0.29+0.46=0.75\)

2. (b) Find the probability that the pupil does neither.

   • \(1-0.75=0.25\)

In a youth group, 48% of members play music, 35% act in plays and 12% do both. A member is chosen at random.

1. (a) Find the probability that the member does exactly one of the two activities.

   • Music only: \(0.48-0.12=0.36\). Plays only: \(0.35-0.12=0.23\)
   • Add: \(0.36+0.23=0.59\)

2. (b) Find the probability that the member does neither.

   • \(P(\text{music or plays})=0.48+0.35-0.12=0.71\)
   • \(1-0.71=0.29\)

A and B are mutually exclusive events with \(P(A)=x\), \(P(B)=3x+0.14\) and \(P(A\cup B)=0.74\).

1. (a) Write an equation for x and solve it.

   • Mutually exclusive: \(P(A\cup B)=P(A)+P(B)\)
   • \(0.74=x+3x+0.14\)
   • \(4x=0.74-0.14=0.6\), so \(x=0.15\)

2. (b) Find P(A) and P(B).

   • \(P(A)=0.15\)
   • \(P(B)=3\times0.15+0.14=0.59\)

A customer at a shop is chosen at random. The Venn diagram shows probabilities for A, the customer pays by card, and B, the customer uses a loyalty card.

1. (a) Find the value of y.

   • All the regions add to 1: \(0.27+0.21+y+0.19=1\)
   • \(y=1-0.67=0.33\)

2. (b) Find P(B).

   • P(B) = both + B only = \(0.21+0.33=0.54\)

3. (c) Use P(A) + P(B) − P(A ∩ B) to find P(A ∪ B), and check it against the regions inside the circles.

   • Formula: \(0.48+0.54-0.21=0.81\)
   • Regions: \(0.27+0.21+0.33=0.81\)