Finding missing probabilities from two or three given ones: at least one, at most one, only one
Statistics Sec 3 Term 1, Unit 3, Lesson 1
GoalFill in all four regions from two or three given probabilities, and answer at least one, only one, at most one and neither.
New words
four regions · A only, both, B only and neither: their probabilities add up to 1exactly one · A only or B only, not bothat most one · not both: everything except the overlap
Explanation
Whenever two events appear, think of four regions: A only, both, B only and neither. Their probabilities always add up to 1, and every phrase of a problem is a sum of some of them, as the table shows. “At least one” is the three regions inside the circles. “Exactly one” is A only plus B only. “At most one” is everything except the overlap, and “neither” is the outside. Be careful: “not both” and “neither” are different events. In a problem, start from the overlap if you know it, subtract it from P(A) and from P(B) to get the two “only” regions, and use the total 1 for the region that is left. If instead you are given P(neither), you get P(A ∪ B) first and then the overlap from P(A) + P(B) − P(A ∪ B).
Phrase
Regions
Probability
A only
A − B
P(A) − P(A ∩ B)
B only
B − A
P(B) − P(A ∩ B)
both
A ∩ B
P(A ∩ B)
at least one
A ∪ B
P(A) + P(B) − P(A ∩ B)
exactly one
(A − B) ∪ (B − A)
P(A ∪ B) − P(A ∩ B)
at most one (not both)
(A ∩ B)′
1 − P(A ∩ B)
neither
(A ∪ B)′
1 − P(A ∪ B)
Worked examples
P(A) = 0.6, P(B) = 0.5 and P(neither) = 0.15. Then P(A ∪ B) = 0.85 and P(A ∩ B) = 0.6 + 0.5 − 0.85 = 0.25. A only: 0.6 − 0.25 = 0.35. B only: 0.5 − 0.25 = 0.25. Exactly one: 0.35 + 0.25 = 0.60. At most one: 1 − 0.25 = 0.75. The regions 0.35, 0.25, 0.25, 0.15 add up to 1.
With fractions: P(A) = 1/2, P(B) = 1/3 and P(A ∩ B) = 1/6. Then P(A ∪ B) = 1/2 + 1/3 − 1/6 = 2/3, so P(neither) = 1/3. A only is 1/2 − 1/6 = 1/3 and B only is 1/3 − 1/6 = 1/6. Exactly one: 1/3 + 1/6 = 1/2. At most one: 1 − 1/6 = 5/6.
Two buses. The probability that bus 1 is on time is 0.7, that bus 2 is on time is 0.6, and that both are on time is 0.45. At least one on time: 0.7 + 0.6 − 0.45 = 0.85. Neither on time: 1 − 0.85 = 0.15. Exactly one: 0.85 − 0.45 = 0.40. Not both on time: 1 − 0.45 = 0.55. Notice that “not both” (0.55) is much larger than “neither” (0.15).
Class activity
Ask 20 classmates two yes/no questions, for example “Do you walk to school?” and “Do you have a brother or a sister in the school?”. Count the four groups, turn the counts into probabilities (divide by 20) and fill the four regions of a Venn picture. Then find the probability of “at least one yes”, “exactly one yes” and “at most one yes”.
Practice
1
P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. Find P(A ∪ B), P(neither) and P(exactly one).
