Accenture Sample Problems Based On Random Events

Below are three problems which are based on the probability of random events (with similar concepts).

Question 1

There are two bags labeled I and II containing roses - 19 white and 30 yellow. If you are allowed to move the flowers between the bags, what will be the maximum probability of getting a white rose from a bag chosen at random?
a)11/16
b)16/21
c)15/18
d)13/16

Answer : a)11/16

Solution:

Part 1 : Let us arrange roses such that the probability of picking a white rose will be maximized
For such problems you can follow a general rule to arrange items:
Consider there are two bags bag A and bag B. Lets say bag A contains P number of a item1 and bag B contains Q number of item1.
Arrangement to maximize the probability of choosing item1 : Put 1 number of item1 in one bag and move remaining (P - 1) item2 to other bag.
Arrangement to minimize the probability of choosing item1 : Put 1 number of item2 in one bag and move remaining (Q - 1) item2 to other bag.
In our case, we have to find arrangement to maximize the probability of choosing a white flower
Therefore, we have to keep 1 white flower in 1 bag(say bag 1) and move remaining 18 white flowers to another bag(say bag 2).

Part 2 : Based on arrangement we made (refer part 1), find the probability of choosing a white flower
In the above scenario, Probability to choose a white flower = Probability of choosing bag I x Probability of choosing white flower from bag I + Probability of choosing bag II x Probability of choosing white flower from bag II ...(1)
Since there are only two bags, the probability of choosing bag I = probability of choosing bag II = 1/2
Probability of choosing white flower from bag I = number of white flowers in bag I / total number of flowers in bag I = 1/1 = 1
Probability of choosing white flower from bag II = number of white flowers in bag II / total number of flowers in bag II = 18/(18+30) = 18/48 = 6/16
Substituting the above values in eq 1, we get.

Probability to choose a white flower = 1/2 x 1 + 1/2 x 6/16 = 11/16

Question 2

Meera is testing Sona's proficiency in probability and poses the following question:" There are two boxes containing 25 pens and 33 pencils of same size. You can move the pencils between the boxes so that when you choose a box at random and a pencil at random from the chosen box the probability of getting a pencil minimized. Then that minimum probability is:" Can you help Sona to find the answer?

a)11/38
b)21/19
c)21/13
d)13/11

Answer : a)11/38

Solution:
This is very similar to first problem except that the condition is reversed. Here we have to find an arrangement such that probability of choosing a pencil at random is minimized. That arrangement would be to put 1 pen in 1st box and 24 pens with 33 pencils in 2nd box.
In above arrangement, probability of choosing a pencil = Probability of choosing box1 x Probability of choosing a pencil from box1 + Probability of choosing box2 x Probability of choosing a pencil from box2
= 1/2 x number of pencils in box1 / total number of items in box1 + 1/2 x number of pencils in box2 / total number of items in box2
= 1/2 x 0/1 + 1/2 x [33/(33+24)] = 0 + 11/38 = 11/38

Hence the answer is 11/38.

Question 3

A fruit seller has 32 oranges and 14 apples in two bags separately and he is allowed to move the oranges and apples between the bags. Now find the Ratio of the maximum probability of picking a bag and an apple from the chosen bag at random to the minimum probability of picking a bag and an orange from the chosen bag at random.
a)25:28
b)18:15
c)29:16
d)25:18

Answer : c)29:16

Solution :

Part :1 To find the maximum probability of picking a bag randomly and an apple from the chosen bag
(As discussed above problem 1)
To maximize the probability of apple, put 1 apple in one bag and move 13 apples to the bag of orange.
Then the required probability 1/2 + [1/2 x 13/45 ] = 1/2 x (1+13/45) = 1/2 x (58/45) = 29/45.

Part :2 To find the minimum probability when picks an orange at random from a chosen bag
(As discussed in problem2)
To minimize the probability of orange, put 1 apple in one bag and 13 apples with 32 orange in second bag.
Then the minimum probability of getting orange = 0 x 1/2 + 1/2 x 32/45 = 16/45

Part : 3 To find the required ratio
Required ratio = maximum probability / minimum probability = (29/45) / (16/45) = 29 / 16

Hence the answer is 29:16.