# probability distributions

1. Find the value of (a) in the following discrete probability distribution:

X -2 0 2

P(x): a 0.35 0.25

2. A binomial distribution is based on n=25 and p=0.1. Find the probability that x =1.

3 A production process produces parts with weights that are normally distributed with a mean of 1.75 ounces and a standard deviation of 0.15 ounces. If specifications call for weights between 1.50 and 2.25 ounces, what percent are not within specifications?

4 If the scores on a test were normally distributed with a mean equal to 500 and a standard deviation equal to 100, what test score would produce a z score equal to 1.5?

5 A certain intersection has a stop light that some residents feel is inappropriately placed. They estimate that 25% of all cars through the intersection do not stop for a red light. If this is true, how many in a sample of 300 cars encountering a red light would one expect to see not stop? What is the probability that less than 20% of the 300 cars would be observed running a red light?

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

1. Sum of Probabilities should be 1.

So a + .35+ .25 = 1

or a = .4

2. P(X=1) = (n C x)*(p^x)*(1-p)^n-x

= 25C1* .1^1*.9^24

= .199

3. Mean(M) = 1.75

Std Dev = .15

Lower Limit = 1.5

Upper Limit = 2.25

Z lower = (1.5 - 1.75)/.15 ( Z is a normal variate given by Z = (X- Mean)/(Std Dev)...here X is our limits.

Z lower = -1.66

Similarly Z upper = (2.25 - 1.75)/.15

= 3.333

So P(1.5<=X<=2.25) = P(-1.66<=Z<=3.33)

From Normal tables we have P= .9511 ( Not this P(Z<=3.33) - P(Z<=-1.66)..so .999 - .0484)

Percent no within specs = 1- .9511 = .0488

4. M= 500

Std Dev = 100

z = 1.5

z = ( x-m)/(std dev)

so 1.5 = (x - 500)/100

so X = 650.

5. p = .25

n = 300

So expected no of cars not stopping = n*p

= 300*.25 = 75

Prob that less than 20% cars , so Prob that that less than 60 cars( 20% of 300) jump the light.

Poisson Distribution P ( X<=60) = .0433 ( Note this is claculated using an inbuilt function called POISSON in Excel.

https://brainmass.com/statistics/probability/distributions-discrete-probability-distribution-6943