alexander have done his homework with 90% probability, mike with 95% and elisa with 75% probability. in what probability
a) at least two students have done their homework
b)no more than two sudents have done their homework
c) none of the students have done their homework
+23254 The probability that Alexander does his homework is 0.90.
This means that the probability that Alexander does not do his homework is 1.00 - 0.90 = 0.10.
The probability that Mike does his homework is 0.95.
This means that the probability that Mike does not do his homework is 1.00 - 0.95 = 0.05.
The probability that Elisa does her homework is 0.75.
This means that the probability that Elisa does not do her homework is 1.00 - 0.75 = 0.25.
I'm first going to find the probability of different situations.
First: What is the probability that all three do their homework?
This occurs only if A does his, M does his, and E does hers.
Since these are assumed to be independent situations, multiply the probabilities together:
0.90 x 0.95 x 0.75 = 0.64125
Second: What is the probability that only two of the three do their homework?
This can happen in three different ways:
1) If A does his, M does his, and E doesn't do hers: 0.90 x 0.95 x 0.25 = 0.21375.
2) If A does his, M doesn't do his, and E does hers: 0.90 x 0.05 x 0.75 = 0.03375.
3) If A doesn't do his, M does his, and E does hers: 0.10 x 0.95 x 0.75 = 0.07125.
To find the total probability for this case, add these three values together: = 0.31875.
Third: What is the probability that only one of the three does his/her homework?
This can happen in three different ways:
1) If A does his, M doesn't do his, and E doesn't do hers: 0.90 x 0.10 x 0.25 = 0.01125.
2) If A doesn't do his, M does his, and E doesn't do hers: 0.10 x 0.95 x 0.25 = 0.02375.
3) If A doesn't do his, M doesn't do his, and E does hers: 0.10 x 0.05 x 0.75 = 0.00375
To find the total probability for this case, add these three values together: = 0.03875
Fourth: What is the probability that none of the three do their homework?
This can happen only when all three do not do their homework: 0.10 x 0.05 x 0.25 = 0.00125
Now, to answer the questions:
1) What is the probabilility that at least two students have done their homework?
This is the probability that either all three have done their homework or exactly two have done
their homework: 0.64125 + 0.31875 = 0.96
2) What is the probability that no more than two students have done their homework?
This is the probability that either none have done their homework or exactly one has done
his/her homework: 0.00125 + 0.03875 = 0.04
You may have noticed that I could have just subtracted the answer from question 1 from 1.00!
3) What is the probability that none of the three do their homework?
This was calculated above; it is 0.00125
