SOCR EduMaterials Activities Discrete Probability examples
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*a. We will denote the number of unbruised peaches in our first selection X, and the number of unbruised peaches in our second selection Y. <math>P(shipment=A)=</math>P(all four peaches are unbruised)+P(three of the peaches are unbruised) X P(four additional peaches are unbruised)= <math>P(X=4)+P(X=3)\times P(Y=4).</math> | *a. We will denote the number of unbruised peaches in our first selection X, and the number of unbruised peaches in our second selection Y. <math>P(shipment=A)=</math>P(all four peaches are unbruised)+P(three of the peaches are unbruised) X P(four additional peaches are unbruised)= <math>P(X=4)+P(X=3)\times P(Y=4).</math> | ||
*<math> X \sim b(4,0.9) </math>, <math> P(X=4)= {4 \choose 4} 0.9^4 0.1^0= 0.6561</math> | *<math> X \sim b(4,0.9) </math>, <math> P(X=4)= {4 \choose 4} 0.9^4 0.1^0= 0.6561</math> | ||
- | *<math> Y \sim b(4,0.9) </math>, <math> P( | + | *<math> Y \sim b(4,0.9) </math>, <math> P(Y=3)= {4 \choose 3} 0.9^3 0.1^1= 0.2916</math> |
- | *<math> Y \sim b(4,0.9) </math>, <math> P( | + | *<math> Y \sim b(4,0.9) </math>, <math> P(Y=4)= {4 \choose 4} 0.9^4 0.1^0= 0.6561</math> |
Therefore, <math>P(shipment=A)=0.6561+ (0.2916)(0.6561)=0.8474 | Therefore, <math>P(shipment=A)=0.6561+ (0.2916)(0.6561)=0.8474 | ||
*b.</math> | *b.</math> |
Revision as of 00:06, 30 April 2007
- Description: You can access the applets for the distributions at http://www.socr.ucla.edu/htmls/SOCR_Distributions.html .
- Example 1:
Find the probability that 3 out of 8 plants will survive a frost, given that any such plant will survive a frost with probability of 0.30. Also, find the probability that at least 1 out of 8 will survive a frost. What is the expected value and standard deviation of the number of plants that survive the frost?
- Answer:
- ,
Below you can see SOCR snapshots for this example:
- Example 2:
If the probabilities of having a male or female offspring are both 0.50, find the probability that a familiy's fifth child is their first son.
- Answer:
- P(X = 5) = 0.505 = 0.03125
- Example 3:
A complex electronic system is built with a certain number of backup components in its subsystem. One subsystem has 4 identical components, each with probability of 0.20 of failing in less than 1000 hours. The subsystem will operate if at least 2 of the 4 components are operating. Assume the components operate independently.
- Answer:
- a. Find the probability that exactly 2 of the 4 components last longer than 1000 hours.
- b. Find the probability that the subsystem operates longer than 1000 hours.
- a. ,
- b.
- Example 4:
- P(X = 5) = 0.740.3 = 0.07203 where X represents the number of trials.
- Example 5:
- Example 6:
- a.
- b. P(X = 0) = .001,0.001 = 0.1nn = 3.
- In the first snapshot below, where which is too small. In the second snapshot, we can see that when n is increased to 3, increases to nearly 1.
- Example 7:
Construct a probability histogram for the binomial probability distribution for each of the following: n=5,p=0.1, n=5,p=0.5, m=5,n=0.9. What do you observe? Explain.
- Answer:
We observe that if p=0.5 the distribution resembles the normal distribution, with mean np = 0.25. Values above and below the mean are distributed symmetrically around the mean. Also, the probability histograms for p=0.1 and p=0.9 are mirror images of each other.
- Example 8:
On a population of consumers, 60% prefer a certain brand of ice cream. If consumers are randomly selected,
- a. what is the probability that exactly 3 people have to be interviewed to encounter the first consumer who prefers this brand of ice cream?
- b. what is the probability that at least 3 people have to be interviewed to encounter the first consumer who prefers this brand of ice cream?
- Answer:
- Now we subtract this from 1 to find the complement, which is the event that at least 3 people WILL have to be interviewed to find the first case:
1 − 0.84 = 0.16
- Example 9:
The alpha marketing research company employs consumer panels to explore preferences
- Answer:
- a. ,
- b. In this part we will look at every group of 5 people as one unit. We will denote success as the event that all 5 people in a group prefer the new product, and failure as the event that at least one person in a group does not prefer the new product. Therefore, p = 0.01024. So,
P(X = 60) = (1 − .01024)60 = 53.9%
- Below you can see a SOCR snapshot for this example:
- Example 10:
- Answer:
- a. We will denote the number of unbruised peaches in our first selection X, and the number of unbruised peaches in our second selection Y. P(shipment = A) = P(all four peaches are unbruised)+P(three of the peaches are unbruised) X P(four additional peaches are unbruised)=
- ,
- ,
- ,
Therefore, P(shipment = A) = 0.6561 + (0.2916)(0.6561) = 0.8474 * b.