Tutorial 1
1.
The compact discs from a certain supplier are analyzed for scratch and shock resistance. The results from 100 discs tested are summarized as follows: Scratch Resistance Hig Low h 30 10 22 8
High Shock Mediu Resistanc m e Low 25 5 Let A denote the event that a disc has high shock resistance, and B denote the event that a disc has high scratch resistance. If sample is selected at random, determine the following probabilities: i. P(A) ii. P(B) iii. P(B’) iv. (AUB) v. P(A B) vi. P(AUB’) vii. P ( A B ) viii. P ( B A) Solution: i. 0.4 ii. 0.77 iii. 0.23 iv. 0.87 v. 0.3 vi. 0.53 vii. 0.39 viii. 0.75 2.
A box contains 3 black and 4 while balls. Two balls are drawn at random one at a time without replacement. i. What is the probability that a second ball drawn is black? ii. What is the conditional probability that first ball drawn is black if the second ball is known to be black? Solution: (Use tree diagram) i. 3/7 ii. 1/3 3. In a group of 22 players of Harimau Muda team, only 7 of them are totally fit for the Asian game tournament and every person of the remaining 15 players has either knee injury or flue fever or both. However 8 of the players have knee injuries and 10 have flu fever. Let A be the event of knee injuries and B be the event of flu fever. If one of them selected randomly from this group, what is the probability that he has i. both, knee injury and high flu fever? ii. knee injury only? iii. flu fever only?
iv. not either knee injury or flu fever? Solution: (Use Venn diagram) i. 3/22 ii. 5/22 iii. 7/22 iv. 7/22 4.
An agricultural research establishment grows vegetables and grades each one as either good or bad for taste, good or bad for its size, and good or bad for its appearance. Overall, 78% of the vegetables have a good taste. However, only 69% of the vegetables have both a good taste and a good size. Also, 5% of the vegetables have a good taste and a good appearance, but a bad size. Finally, 84% of the vegetables have either a good size or a good appearance. i. If a vegetable has a good taste, what is the probability that it also has a good size? ii. If a vegetable has a bad size and a bad appearance, what is the probability that it has a good taste? Solution: (Use Venn diagram) i. 0.89 ii. 0.25
5.
Three machines M1, M2 and M3 produce identical items of their respective output 5%, 4% and 3% of the items are faulty. On a certain day, M1 has produced 25%, M2 has produced 30% and M3 has produced 45% of the total output. An item selected at random is found to be faulty. What are the chances that it was produced by M3? Solution: (Use tree diagram) P(M3 given F) = 0.36
6.
Suppose that a test for Influenza A, H1N1 disease has a very high success rate: if a tested patient has the disease, the test accurately reports this, a ’positive’, 99% of the time, and if a tested patient does not have the disease, the test accurately reports that, a ’negative’, 95% of the time. Suppose also, however, that only 0.1% of the population have that disease. i. What is the probability that the test returns a positive result? ii. If the patient has a positive, what is the probability that he has the disease? iii. What is the probability of a false positive? Solution: (Use tree diagram) Let D be the event that the patient has the disease, and E is the event that the test returns a positive result.
i. ii. iii.
P(E) = 0.05094 P(D given E) = 0.019 P( False positive) = 1 – P(D given E) = 0.981 oooOOOooo
