Download Binomial Probability Distribution

Binomial Probability Distribution
A2.S.15
A2.S.16
Know and apply the binomial probability formula to events involving the terms exactly, at least, and
at most
Use the normal distribution as an approximation for binomial probabilities
If p represents the probability of a success in each of n independent trials, then
The probability of r successes in n trials where p is the probability of success and q = 1 – p (the probability of failure)
P(r successes in n trials) = n Cr p r q nr
Ex: Find the probability of obtaining exactly 3 heads in 5 tosses of a fair coin.
n = 5 because there are 5 trials (5 total tosses)
r = 3 because we want 3 heads (3 successes)
p = 0.5 because the probability of getting a success (heads) is 1 out of 2.
q = 0.5 because the probability of not getting a heads (getting a tails) is also 1 out of 2.
So,
P(3 H in 5 tosses) = 5 C 3 0.53 0.52
= 10  0.125  0.25
= 0.3125
The probability of obtaining at least r successes in n trials is the sum of the probabilities of obtaining r successes, r + 1
successes, …, continuing up to n successes.
3
Ex: In a baseball game, the probability that John will get on base safely when he comes to bat is . What is the probability
5
that he will get on base safely at least 3 out of 4 times he comes to bat?
n = 4 because there are 4 trials
r = 3 and 4 because at least 3 means 3 or more
3
p  because it represents the probability of getting on base safely
5
3 2
q  1   because it represents the probability of not getting on base safely
5 5
3
1
4
3 2
3 2
P(getting on base safely in at least 3 out of 4 at bats) = 4 C 3      4 C 4    
5 5
5 5
216 81 297


=
625 625 625
0
The probability of obtaining at most r successes in n trials is the sum of the probabilities of obtaining 0 successes, 1
success, …, continuing up to r successes.
1
Ex: In the month of February at a ski resort, the probability of snow on any day is . What is the probability that snow will
4
fall on at most 2 days of a weeklong trip to that ski resort in February?
n = 7 because there are 7 days in a weeklong trip
r = 0, 1, and 2 because at most 2 means 2 days or less.
p = ¼ because it is the probability of success
q = ¾ because it is the probability of failure, that is, the probability of no snow on a given day
0
7
1
6
2
 1  3
 1  3
 1  3
P(getting at most 2 days of snow) = 7 C 0      7 C1      7 C 2    
4 4
4 4
4 4
2187
1 729
1 243
 7 
 21  
= 1 1
16384
4 4096
16 1024
2187 5103 5103 12393



 0.7564
=
16384 16384 16384 16384
5
Normal Approximation to the Binomial Distribution
When n is large, there are many cases of success to be found for questions on multiple successes (e.g., at least, at most).
These binomial probabilities can be quite tedious to calculate. That is where the normal distribution comes in. The normal
distribution allows us to approximate binomial probabilities when n is large and p is close to 0.5. To find the normal
approximation to the binomial distribution, we must first find the mean and standard deviation of the normal distribution.
We use   np or x  np to find the mean.
We use   np1  p to find the standard deviation.
Where n is the number of trials of the binomial distribution and p is the probability of success.
We must also adjust the endpoints of the interval by 0.5 to either include or exclude the endpoints (this is called continuity
correction). The normalcdf function can then be used to find the probability. To use normalcdf, hit 2 nd, VARS for the
Distribution menu. Choose choice 2. The syntax for normalcdf is as follows:
Normalcdf(lower value, upper value, mean, standard deviation)
The normal distribution is considered a good approximation to the binomial distribution when np  10 and n1  p   10 .
Ex. A department store has determined that the probability of a customer making a purchase is approximately 0.6. Find the
probability, to the nearest thousandth, that at least 70 of the next 100 customers will make a purchase.
Solution: The mean is np = (100)(0.6) = 60 and the standard deviation is np1  p  100 0.60.4  24  4.899 .
The normal distribution with mean 60 and standard deviation 4.899 will give a good approximatation.
The probability of at least 70 out of 100 customers making a purchase is approximately equal to the area under this normal
curve from 69.5 and 100.5. Notice that I subtracted 0.5 from 70 to include 70 in the interval and I added 0.5 to 100 to
include 100 in the interval.
Now enter on the homescreen: normalcdf(69.5. 100.5, 60, 4.899)
You will get 0.0262396804.
The probability that at least 70 customers out of 100 will make a purchase is about 0.026.
Name _______________________________________
Algebra II & Trigonometry
1. If the probability of winning a game is
27
125
54
(2)
625
(1)
Date ___________________________________
Probability
3
, then the probability of winning exactly 3 games out of 4 played is
5
216
(3)
625
532
(4)
625
2. When Nick plays cards with Lisa, the probability that Nick will win is
6
. If they play three games of cards and there are
10
no ties, what is the probability that Lisa will win all three games?
3. If the probability of hitting a target is
3
256
12
(2)
256
(1)
3
, then the probability of hitting the target exactly once in four tries is
4
27
(3)
256
36
(4)
256
4. In a family of six children, what is the probability that exactly one child is female?
6
32
(1)
(3)
64
64
7
58
(2)
(4)
64
64
5. Each day the probability of rain on a tropical island is
7
. Which expression represents the probability that it will rain on
8
the island exactly n days in the next 3 days?
n
(1)
 7  1
3 Cn    
8 8
3
7  1
(2) 3 C 3    
8 8
3 n
3
(3)
 7  1
n C3    
8 8
(4)
8 C7
n
n
3n 38n
6. In basketball, Nicole makes 4 baskets for every 10 shots. If she takes 3 shots, what is the probability that exactly 2 of
them will be baskets?
(1) 0.288
(3) 0.600
(2) 0.432
(4) 0.960
7. The probability of Gordon’s team winning any given game in a 5-game series is 0.3. What is the probability that
Gordon’s team will win exactly 2 games in the series?
(1) (0.3)²(0.7)³
(3) 10(0.3)²(0.7)³
(2) 5(0.3)³(0.7)²
(4) 5(0.3)²(0.7)
8. Gordon tosses a fair die six times. What is the probability that he will toss exactly two 5’s?
2
4
2
4
 5  1
(1) 6 C 5    
6 6
(2)
5  1
6 C2    
6 6
2
4
2
4
 1  5
(3) 6 C 5    
6 6
(4)
 1 5
6 C2    
6 6
9. If three fair coins are tossed, what is the probability of getting at least two heads?
1
1
(1)
(3)
8
2
2
3
(2)
(4)
8
3
10. If a fair coin is tossed five times, what is the probability of tossing exactly three heads?
11. A spinner is divided into five equal sectors labeled 1 through 5. What is the probability of getting at most two prime
numbers in three spins?
12. A coin is biased so that the probability of obtaining heads is
3
. What is the probability of obtaining at least three heads
5
in four tosses of the coin?
3
. If they are playing a three-game series
4
this weekend, what is the probability that the Mets will win at least 2 out of 3 games?
13. The probability of the Mets winning a game against the Diamondbacks is
14. A traffic light on Hempstead Turnpike is green for 40 seconds, yellow for 5 seconds, and red for 15 seconds out of
every minute. What is the probability that at least four of the next 5 cars get a green light?
15. At a university, the probability that an incoming freshman will graduate within four years is 0.553. Use the normal
approximation to estimate, to the nearest thousandth, the probability that
(a) at least 60 out of a group of 100 incoming freshman will graduate in four years.
(b) no more than 80 out of a group of 150 incoming freshman will graduate in four years.