5. There are two groups of subjects one of which contains 5 science subjects and 3 engineering subjects and the other consists of 3 science and 5 engineering subjects. An unbiased die is cast. If the number 3 or number 5 turns up, a subject is selected at random from the first group, otherwise the subject is selected at random from the second group. Find the probability that an engineering subject is selected ultimately.

To find the probability that an engineering subject is selected ultimately, we need to consider two cases:

Case 1: Number 3 or 5 turns up (selected from the first group):
The probability of rolling a 3 or 5 on an unbiased die is 2/6 = 1/3.
Out of the first group, there are a total of 8 subjects (5 science + 3 engineering).
The probability of selecting an engineering subject from the first group is 3/8.

Case 2: Any number other than 3 or 5 turns up (selected from the second group):
The probability of rolling a number other than 3 or 5 on an unbiased die is 4/6 = 2/3.
Out of the second group, there are a total of 8 subjects (3 science + 5 engineering).
The probability of selecting an engineering subject from the second group is 5/8.

Now, we can calculate the overall probability of selecting an engineering subject ultimately by considering both cases:

Probability = (Probability of Case 1) + (Probability of Case 2)
= (1/3) * (3/8) + (2/3) * (5/8)
= 3/24 + 10/24
= 13/24

Therefore, the probability that an engineering subject is selected ultimately is 13/24.

To find the probability that an engineering subject is selected ultimately, we need to consider all the possible outcomes and calculate the probability for each outcome.

Let's denote the event of selecting an engineering subject as E and the event of selecting a science subject as S.

First, let's consider the possible outcomes when a number 3 or 5 turns up on the die:
- If number 3 or number 5 turns up, a subject is selected at random from the first group (5 science and 3 engineering subjects). The probability of selecting an engineering subject from the first group is 3/8.
- The probability of selecting a science subject from the first group is 5/8.

Now, let's consider the possible outcomes when any number other than 3 or 5 turns up on the die:
- If any number other than 3 or 5 turns up, a subject is selected at random from the second group (3 science and 5 engineering subjects). The probability of selecting an engineering subject from the second group is 5/8.
- The probability of selecting a science subject from the second group is 3/8.

To find the ultimate probability of selecting an engineering subject, we need to consider the probability of each outcome and weigh it by the probabilities of the die results. Since the die is unbiased, each number has an equal probability of 1/6.

The probability of getting a number 3 or 5 is 2/6 = 1/3.
The probability of getting any other number is 4/6 = 2/3.

Using these probabilities, we can calculate the ultimate probability of selecting an engineering subject:

Probability of selecting an engineering subject ultimately = (Probability of selecting an engineering subject from the first group * Probability of getting a number 3 or 5 on the die) + (Probability of selecting an engineering subject from the second group * Probability of getting any other number on the die)

= (3/8 * 1/3) + (5/8 * 2/3)
= 1/8 + 10/24
= 1/8 + 5/12
= 3/24 + 10/24
= 13/24

Therefore, the probability that an engineering subject is selected ultimately is 13/24.

Engineering -- E

Science ------ S

Prob (E is picked from 1st group) = 3/8
prob(E is picked from the 2nd group) = 5/8

first picking is determined by a die toss of 3 or 5
which has a prob of 2/6 or 1/3
2nd picking is determined by any other die toss
which is 2/3

prob of your stated event
= (1/3)(3/8) + (2/3)(5/8)
= 3/24 + 10/24
= 13/24