Math 125 Statistics Assignment: Probability Problems and Solutions

This assignment solution addresses several probability and statistics problems. The first problem calculates the probability of at least one defective item given a 2% defect rate. The second problem determines the probability of a baseball player getting at least 5 hits given his batting average. The solution then explores normal distribution problems, calculating probabilities based on given values and finding probabilities greater or less than certain values. The assignment also involves calculating probabilities for a soda bottle amount and finding values based on standard deviations from the mean of a normal distribution. All solutions are presented with detailed calculations and explanations.

Math *125*009 (STATISTICS)
MUST SHOW ALL WORK
1. A manufacturing machine has a 2% defect rate.
If 3 items are chosen at random, what is the probability that at least one
will have a defect?
let P be the probability of defective item and
p=0.02
q=1-0.02 = 0.98
P(at least 1defective)= 1 – P(Non-defective)
P( Non-defective) = 0.983
P(at least 1defective)= 1 – 0.983 = 0.0588
2. A high School baseball player has a 0.287 batting average. In one game,
he gets 9 at bats. What is the probability he will get at least 5 hits in the
game?
P= 0.287
P(x)=(n
x )Px (1−P)n−x
P(x=5) = (9
5 )0.2875 (1−0.287)9−5
= 0.0634
P(x=6)= (9
6 )0.2876 (1−0.287)9−6
= 0.0170
P(x=7)= (9
7 )0.2877 (1−0.287)9−7
= 0.00293
P(x=8)= (9
8 )0.2878 (1−0.287)9−8
= 0.00029
P(x=9)= (9
9 )0.2879 (1−0.287)9−9
= 0.00001
P( atleast 5 hits) = 0.0634+0.0170+0.00293+0.00029+0.00001
=0.08363

3. A continuous random variable is normally distributed. The probability
that a value in the distribution is greater than 21 is 0.6658. Find the
probability that a value in the distribution is less than 21?
P( less than 21) = 1- P(greater than 21)
= 1- 0.6658
=0.3342
4. A continuous random variable is normally distributed. The probability
that a value in the distribution is less than 171 is 0.0051. Find the
probability that a value in the distribution is greater than 171?
P( greater than 171) = 1- P(less than 171)
= 1- 0.0051
=0.9949
5. The amount of soda in a randomly selected bottle varies according to a
normal distribution with mean 17 ounces and standard deviation 5.9
ounces. What is the probability that a randomly selected bottle will be
less than 17 ounces? Enter your answer as a decimal?
P(x<17)
z=17−17
5.9 = 0
Probability of (z=0) = 0.5 from standard normal table
P(x<17) =0.5
6. A random variable is normally distributed with mean 38.9 and standard
deviation 7.6. What value is 2 standard deviations below the mean?
z= x−38.9
7.6 =2
x−38.9
7.6 =2
x= 2(7.6) +38.9 =54.1
The value is 54.1
7. A random variable is normally distributed with mean 93.7 and standard
deviation 12.8. What value is 3 standard deviations above the mean?
z= x−93.7
12.8 =3
x−93.7
12.8 =3
x= 3(12.8) +93.7 =132.1

The value is 132.1