Question

1. A password of length 5 is generated at random as a sequence of letters from A,...,Z with repetitions allowed. What is the probability that in every pair of adjacent positions in the password, the letters are different (for example, XJXTQ but not XXJTQ) ? 2. Bowl #1 contains 2 red chips and 3 white chips. Bowl #2 contains 3 red chips and 4 white chips. One of the bowls is chosen at random, and a chip is withdrawn from that bowl. (a) What is probability that the withdrawn chip is red? (b) Given that the withdrawn chip is red, what is the probability that bowl #1 was chosen? 3. A manufacturer has three plants that produce the same model of a device being marketed to the public. Plant #1 produces 50% of the devices, plant #2 produces 30% of the devices, and plant #3 produces 20% of the devices. As a percentage of its total output, plant #1 produces 1% defectives; the corresponding figures for plants #2 and #3 are 2% and 3% respectively. (a) What is the probability that a purchased device is defective? (b) If a purchased device is found to be defective, what is the probability that it was produced at plant #1? 4. A bowl contains 2 red chips and 2 white chips. A chip is withdrawn, its color noted, and then replaced in the bowl together with another chip of the same color. The bowl now contains 5 chips. Another chip is then withdrawn. (a) What is the probability that the second withdrawn chip is red? (b) Given that the second withdrawn chip is red, what is the probability that the first withdrawn chip was red? 5. A bowl contains 3 red chips and 1 white chip. Two chips are withdrawn simultaneously and discarded. Then a third chip is withdrawn. (a) What is the probability that the third withdrawn chip is red? (b) Given that the third withdrawn chip is red, what is the probability that the first two withdrawn chips were of different colors?

Answer 1

2.a) Probability of choosing any bowl is 1/2
In Bowl 1 probability of choosing red is 2/5
In Bowl 2 probability of choosing red is 3/7
Hence probability of choosing red is (1/2 * 2/5) + (1/2 * 3/7) = 29/70

b) P(choosing bowl 1 | Chip is red) = (P(choose bowl 1 ∩ chip is red))/P(Chip is red) = (1/2 * 2/5)/(29/70) = 14/29

3.a) P(purchased device is defective)= (50/100 * 1/100) + (30/100 * 2/100) + (20/100 * 3/100) = 17/1000

b) P(plant 1 | defective)= P(plant 1 ∩ defective)/P(defective) = (50/10000)/(17/1000) = 5/17

4.a) After 1st withdrawn: condition 1(probability= 1/2): 3 red 2 white. P(2nd withdrawn is red)= 3/5
Condition 2(probability= 1/2): 2 red 3 white. P(2nd withdrawn is red)= 2/5
P(2nd withdrawn is red)= (1/2 * 3/5)+(1/2 * 2/5) =1/2

b) P(1st withdrawn red | 2nd withdrawn red)= P(1st red ∩ 2nd red)/P(2nd red) = (1/2 * 3/5)/(1/2) = 3/5

5.a) If 1st two chips are red (probability= 3C2 / 4C2 = 1/2), then probability of 3rd chip (1 red and 1 green remains) red is 1/2
if 1st 2 chips are of diff colours (probability= (3C1 * 1C1)/ 4C2 = 1/2), then probability of 3rd chip (2 red remains) red is 1
P(3rd is red)= (1/2 * 1/2) + (1/2 * 1) = 3/4
b) P(1st two off diff colour | 3rd red) = P(1st two off diff colour ∩ 3rd red)/ P(3rd red) = (1/2 * 1)/ (3/4) = 2/3