Combined and mutually exclusive events Problem set A One question at a time

Find P(A ∩ B) when:

1. (a) P(A) = 0.55, P(B) = 0.6, P(A ∪ B) = 0.8.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(0.8=0.55+0.6-P(A\cap B)\)
   • \(P(A\cap B)=0.55+0.6-0.8=0.35\)

2. (b) P(A) = 0.45, P(B) = 0.3, P(A ∪ B) = 0.6.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(0.6=0.45+0.3-P(A\cap B)\)
   • \(P(A\cap B)=0.45+0.3-0.6=0.15\)

Find P(A ∪ B) when:

1. (a) P(A) = 0.4, P(B) = 0.35, P(A ∩ B) = 0.15.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(P(A\cup B)=0.4+0.35-0.15=0.6\)

2. (b) P(A) = 0.7, P(B) = 0.45, P(A ∩ B) = 0.3.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(P(A\cup B)=0.7+0.45-0.3=0.85\)

Find P(B) when:

1. (a) P(A) = 0.5, P(A ∩ B) = 0.2, P(A ∪ B) = 0.75.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(0.75=0.5+P(B)-0.2\)
   • \(P(B)=0.75-0.5+0.2=0.45\)

2. (b) P(A) = 0.35, P(A ∩ B) = 0.1, P(A ∪ B) = 0.6.

   • \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
   • \(0.6=0.35+P(B)-0.1\)
   • \(P(B)=0.6-0.35+0.1=0.35\)

Events A and B are mutually exclusive.

1. (a) P(A) = 0.28 and P(A ∪ B) = 0.7. Find P(B).

   • Mutually exclusive: \(P(A\cap B)=0\), so \(P(A\cup B)=P(A)+P(B)\)
   • \(0.7=0.28+P(B)\)
   • \(P(B)=0.7-0.28=0.42\)

2. (b) P(A) = 1/6 and P(B) = 3/8. Find P(A ∪ B).

   • \(P(A\cup B)=P(A)+P(B)\)
   • \(P(A\cup B)=\frac{1}{6}+\frac{3}{8}\)
   • \(=\frac{13}{24}\)

A and B are two events. P(A) = 0.35, P(B) = 0.4 and P(A ∪ B) = 0.7. Are A and B mutually exclusive? Give a reason.

• \(P(A\cap B)=0.35+0.4-0.7=0.05\)
• It is not 0, so A and B can happen together: they are not mutually exclusive.

In a town, 34% of households have a cat, 27% have a dog and 9% have both. A household is chosen at random.

1. (a) Find the probability that the household has a cat or a dog (or both).

   • \(P(\text{either})=0.34+0.27-0.09=0.52\)

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

   • Neither is everything outside the union: \(1-0.52=0.48\)

At a school, a student is chosen at random. The probability that the student plays netball is 0.46, and the probability that the student plays hockey is 0.38. The probability that the student plays netball or hockey (or both) is 0.7.

1. (a) Find the probability that the student plays both sports.

   • \(0.7=0.46+0.38-P(\text{both})\)
   • \(P(\text{both})=0.46+0.38-0.7=0.14\)

2. (b) Find the probability that the student plays netball but not hockey.

   • Netball only = 0.46 − 0.14 = 0.32

A commuter chooses one way to travel to work. The probability of cycling is 0.34 and the probability of taking the train is 0.41. The two are mutually exclusive.

1. (a) Find the probability that the commuter cycles or takes the train.

   • Mutually exclusive, so add the probabilities.
   • \(0.34+0.41=0.75\)

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

   • \(1-0.75=0.25\)

At a college, 52% of students do swimming, 38% do chess and 15% do both. A student is chosen at random.

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

   • Swimming only: \(0.52-0.15=0.37\). Chess only: \(0.38-0.15=0.23\)
   • Add: \(0.37+0.23=0.6\)

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

   • \(P(\text{swimming or chess})=0.52+0.38-0.15=0.75\)
   • \(1-0.75=0.25\)

A and B are mutually exclusive events with \(P(A)=x\), \(P(B)=2x+0.06\) and \(P(A\cup B)=0.69\).

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

   • Mutually exclusive: \(P(A\cup B)=P(A)+P(B)\)
   • \(0.69=x+2x+0.06\)
   • \(3x=0.69-0.06=0.63\), so \(x=0.21\)

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

   • \(P(A)=0.21\)
   • \(P(B)=2\times0.21+0.06=0.48\)

A student at a school is chosen at random. The Venn diagram shows probabilities for A, the student takes art, and B, the student takes music.

1. (a) Find the value of y.

   • All the regions add to 1: \(0.32+0.18+y+0.25=1\)
   • \(y=1-0.75=0.25\)

2. (b) Find P(B).

   • P(B) = both + B only = \(0.18+0.25=0.43\)

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: \(0.5+0.43-0.18=0.75\)
   • Regions: \(0.32+0.18+0.25=0.75\)