AI SL · 13.1
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Combined events and mutually exclusive events
13.1.5
Type P(A), P(B) and P(A ∩ B). The four regions fill in. Choose what to shade and read the working. The circles are not drawn to scale.
Type three probabilities to fill the diagram.
A and B are events with P(A) = 0.55, P(B) = 0.35 and P(A ∪ B) = 0.7. Find P(A ∩ B).
Have a go first. Then open the working one line at a time.
\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Adding P(A) and P(B) counts the overlap twice, so one copy is taken off.
\(0.7=0.55+0.35-P(A\cap B)\)
Write the equation before you rearrange it.
\(P(A\cap B)=0.55+0.35-0.7=0.2\)
Swap the unknown with P(A ∪ B).
P(A) = 0.45, P(B) = 0.3 and P(A ∩ B) = 0.1. Find the probability that A or B or both happen.
Have a go first. Then open the working one line at a time.
\(P(A\text{ or }B)=P(A\cup B)\)
'Or' includes both events happening.
\(P(A\cup B)=0.45+0.3-0.1\)
Add, then take off the overlap once.
\(P(A\cup B)=0.65\)
Check: it must be at least as big as P(A) and as P(B).
P(A) = 0.5, P(A ∩ B) = 0.15 and P(A ∪ B) = 0.8. Find P(B).
Have a go first. Then open the working one line at a time.
\(0.8=0.5+P(B)-0.15\)
Put the three known values in first.
\(0.8=0.35+P(B)\)
0.5 − 0.15 = 0.35.
\(P(B)=0.8-0.35=0.45\)
A probability must be between 0 and 1. It is.
A and B are mutually exclusive. P(A) = 0.35 and P(A ∪ B) = 0.6. Find P(B).
Have a go first. Then open the working one line at a time.
\(P(A\cap B)=0\)
They cannot happen together, so the circles do not overlap.
\(P(A\cup B)=P(A)+P(B)\), so \(0.6=0.35+P(B)\)
The intersection term is 0.
\(P(B)=0.6-0.35=0.25\)
A and B are mutually exclusive, with P(A) = 1/4 and P(B) = 2/5. Find P(A ∪ B).
Have a go first. Then open the working one line at a time.
\(P(A\cup B)=\frac{1}{4}+\frac{2}{5}\)
Mutually exclusive: nothing to subtract.
\(=\frac{5}{20}+\frac{8}{20}\)
The lowest common multiple of 4 and 5 is 20.
\(=\frac{13}{20}=0.65\)
Fractions in lowest terms are fine; a decimal is too.
For events A and B, P(A) = 0.6, P(B) = 0.45 and P(A ∪ B) = 0.8. Find the probability that exactly one of A and B happens.
Have a go first. Then open the working one line at a time.
\(P(A\cap B)=0.6+0.45-0.8=0.25\)
Rearrange the combined events formula.
A only: \(0.6-0.25=0.35\). B only: \(0.45-0.25=0.2\)
Take the overlap out of each circle.
\(0.35+0.2=0.55\)
Same as \(P(A\cup B)-P(A\cap B)=0.8-0.25\).