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Binomial Probability
'At Most' and 'At Least'
MathBitsNotebook.com

Topical Outline Algebra 2 Outline MathBits' Teacher Resources
Terms of Use Contact Person:Donna Roberts

When computing 'at least' and 'at most' probabilities, it is necessary to consider,
in addition to the given probability,
• all probabilities larger than the given probability ('at least')
• all probabilities smaller than the given probability ('at most')

The probability of an event, p,
occurring exactlyr times:

n = number of trials
r = number of specific events you wish to obtain
p = probability that the event will occur
q = probability that the event will not occur
(q = 1 - p, the complement of the event)

Illustration:
A bag contains 6 red Bingo chips, 4 blue Bingo chips, and 7 white Bingo chips. What is the probability of drawing a red Bingo chip at least 3 out of 5 times? Round answer to the nearest hundredth.

To solve this problem, we need to find the probabilities that r could be 3 or 4 or 5,
to satisfy the condition 'at least'.
It will be necessary to compute
for r = 3, r = 4 and r = 5 (which means do the formula three times)
and add these three probabilities for the final answer.
Written in summation notation, we need to compute:

For r = 3:
For r = 4:
For r = 5:
Sum:	0.184 + 0.050 + 0.005 = 0.239 rounded to the nearest hundredth = 0.24 ANS

Note: it may be helpful to remember in this problem that
'at most 2 successes' is the compliment of 'at least 3 successes'.

In the following examples, answers will be rounded to the nearest hundredth.

A family consists of 3 children. What is the probability that at most 2 of the children are boys?

Solution:
'At most' 2 boys implies that there could be 0, 1, or 2 boys. The probability of a boy child (or a girl child) is 1/2.

For r = 0:
For r = 1:
For r = 2:
Sum:	.125 + .375 + .375 = .875 rounded to the nearest hundredth = 0.88 ANS

Team A and Team B are playing in a league. They will play each other five times. If the probability that team A wins a game is 1/3, what is the probability that team A will win at least three of the five games?

Solution:
'At least' 3 wins implies 3, 4, or 5 wins.

For r = 3:
For r = 4:
For r = 5:
Sum:	rounded to the nearest hundredth = 0.21 ANS

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As shown in the accompanying diagram, a circular target with a radius of 9 inches has a bull's-eye that has a radius of 3 inches. If five arrows randomly hit the target, what is the probability that at least four hit the bull's-eye? Express answer to the nearest thousandth.

Solution:
'At least' 4 hits implies 4 or 5 hits. The area of the bull's-eye is 9π and the area of the entire target is 81π. The probability of hitting the desired bull's-eye is 1/9.

For r = 4:
For r = 5:
Sum:	rounded to the nearest thousandth = 0.001 ANS

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At Least
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