There are certain basic formulae that are required for solving questions on Probability. They are as follows:

Probability of an event happening =

The probability of an event will range from 0 to 1 that is 0 ≤ P (E) ≤ 1

P (event occurring) + P (event not occurring) = 1

Let us understand the use of these formulae by solving some questions.

**1. A problem is given to Alan, Bob and Carl and their probability of solving it is1/2, 1/3 and 2/5 respectively. What is the probability that none of them solves the problem?**

a) 1/5

b) 1/3

c) 1/2

d) 3/4

e) 11/12

Now the question asks for the probability of the problem not being solved by all three students.

P (Alan not solving AND Bob not solving AND Carl not solving) is what is asked.

It is given that P (Alan solving) = 1/2 , P (Bob solving) = 1/3 and P (Carl solving) = 2/5

P (Alan solving)

= 1/2

P (Alan not solving) = 1- P (Alan solving)

= 1 – 1/2 = 1/2

P (Bob solving) = 1/3

P (Bob not solving) = 1-1/3

= 2/3

P (Carl solving) =2/5

P (Carl not solving) = 1-2/5 = 3/5

Therefore P (Alan not solving AND Bob not solving AND Carl not solving)

= 1/2 x 2/3 x 3/5

= 1/5

The point to be noted here is that when we use ‘AND’ we multiply.

**The correct answer choice is A. **

Now let us look at one more question.

2. **Annie and Chris write the GRE on the same day. The probability of Annie getting a 170 in Math is 1/7 and the probability of Chris getting a 170 in Math is1/5. What is the probability that only one of them get 170 in Math?**

- 1/7
- 2/7
- 1/2
- 3/4
- 4/5

In this question we have to find the P (Exactly one of them getting 170).

So P (Exactly one of them getting 170) can be written as:

P (Annie getting a 170 in Math AND Chris not getting a 170 in Math) OR

P (Chris getting a 170 in Math AND Annie not getting a 170 in Math)

Here we see the use of both ‘AND’ and ‘OR’.

For ‘AND’ we multiply and for ‘OR’ we add.

Now the P (Annie getting a 170 in Math) = 1/7

P (Annie not getting a 170 in Math) = 6/7

P (Chris getting a 170 in Math) = 1/5

P (Chris not getting a 170 in Math) = 4/5

P (Annie getting a 170 in Math AND Chris not getting a 170 in Math) = 1/7 x 4/5

= 4/35

P (Chris getting a 170 in Math AND Annie not getting a 170 in Math) = 1/5 x 6/7

= 6/35

**Putting both together we have**:

P (Annie getting a 170 in Math AND Chris not getting a 170 in Math) OR

P (Chris getting a 170 in Math AND Annie not getting a 170 in Math)

= 4/35+6/35

= 10/35

= 2/7

**The correct answer choice is B.**

*In our next blog on Probability, we will be seeing more questions on independent and dependent events.*