Question 476924: 36% of college students say they use credit card because of the rewards program . You randomly select 10 college students and ask each to name the reason he or she uses credit cards. Find the probability that number of college students who say they use credit cards because of there wards program is (a) exactly two, (b) more than two, and (c) between two and five inclusive. If convenient, use technology to find the probabilities.
(a)P(2)=____
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the probability that any student will use a credit card because of the rewards program is .36
that is 36% divided by 100%.
the probability that a student will use a credit card for other than the rewards program is equal to 1 - .36 = .64
you ask 10 students if they use a credit card.
the probability that exactly 2 students will use a credit card because of the rewards program.
the formula for determining exactly x number of students would be:
p(x) = .36^x * .64^(10-x) * 10Cx
these are the total probabilities as far as i can see them.
p(c) is the probability of using a credit card because of the rewards program.
p(o) is the probability of using a credit card for other reasons.
x is the number of people out of 10 who are using the credit card because of the rewards program.
(10-x) is the number of people out of 10 who are using the credit card for reasons other than the rewards program.
p(x) is the probability of exactly x people out of 10 of using the credit card for the rewards program.
x p(c)^x p(o)^(10-x) 10Cx p(x)
0 1 0.011529215 1 0.011529215
1 0.36 0.018014399 10 0.064851835
2 0.1296 0.028147498 45 0.164156206
3 0.046656 0.043980465 120 0.24623431
4 0.01679616 0.068719477 210 0.242386899
5 0.006046618 0.107374182 252 0.163611157
6 0.002176782 0.16777216 210 0.07669273
7 0.000783642 0.262144 120 0.024651235
8 0.000282111 0.4096 45 0.00519987
9 0.00010156 0.64 10 0.000649984
10 3.65616E-05 1 1 3.65616E-05
sum of all probabilities >>>>>>>>>>>>>>>>>>>>>> 1
the answers to your questions are:
a.) p(2) = 0.164156206
b.) p(>2) = 0.759462744
c.) p([2,5]) = 0.816388571
p(2) is the probability of getting exactly 2 out of 10 people who are using the credit card for the rewards program.
this is just grabbing the p(2) out of the table above because each of those entries is the probability of getting exactly that number of people out of 10 who are using the credit card because of the rewards program.
p(>2) is the probability of getting more than 2 out of 10 people who are using the credit card for the rewards program.
this can be calculated by taking the sum of the probability of (3,4,5,6,7,8,9,10) people out of 10, or of taking the sum of the probability of (0,1,2) people out of 10 and subtracting that from 1.
p([2,5]) is the probability of (2,3,4,5) people out of 10 who are using the credit card because of the rewards program.
the sum of all probabilities is equal to 1 as it should be.
10Cx is the combination formula.
the total probability of exactly x people out of 10 using the credit card becsuse of the rewards program is the probability times the number of possible ways that probability can be incurred.
example:
the probability of getting exactly 1 person out of 10 would be:
.36^1 * .64^9 * 10C1 which would be equal to:
.36^1 * .64^9 * 10
this would be:
c---------
-c--------
--c-------
---c------
----c-----
-----c----
------c---
-------c--
--------c-
---------c
what this is saying is:
it could be the first person or it could be the second person or it could be the third person, etc.
the probability of each of these occurrences is .36^1 * .64^9.
since there are 10 possible ways that can happen, the total probability for 1 person out of 10 would be 10 * .36^1 * .64^9
with 2 people out of 10, it can happen 45 ways.
with 3 people out of 10, it can happen 120 ways.
etc.
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