Binomial Probability
"At Least / At Most"
 Topic Index | Algebra2/Trig Index | Regents Exam Prep Center

 

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 exactly r  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)


Alternative formula form

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 statisfy the condition "at least".
It will be necessary to compute
for r = 3, r = 4 and r = 5 and add these three probabilities for the final answer.
We need to compute:

For r = 3:
For r = 4:
For r = 5:
Sum: .184 + .050 + .005 = .239
rounded to the nearest hundredth = 0.24  ANSWER

 

It may be helpful, in certain problems, to remember that:

 

 

Examples: (answers rounded to the nearest hundredth)

1.  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  ANSWER

 

 
2. 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  ANSWER

 

 
3.  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 and the area of the entire target is .  The probability of hitting the target is 1/9.
For r = 4:
For r = 5:
Sum: rounded to the nearest thousandth = 0.001  ANSWER

 

How to use your TI-83+/84+ graphing calculator with Bernoulli Trials.
Click calculator.

How to use your TI-83+/84+ graphing calculator with At Most/At Least.
Click calculator.