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# Probability of Simple Events

A simple event is an event where all possible outcomes are equally likely to occur. So the probability of simple events will have all possible outcomes equally likely to happen or occur.

For example, when you toss a coin, there are two possible outcomes – heads or tails, and the probability of heads or tails is equal. Similarly, when you roll a die, you can get any of the 6 numbers – 1 to 6, and the chance of any one of these 6 numbers is equal to the others.

A set of all possible outcomes of an event is called sample space, usually represented as ‘S’ with curly braces { }. So with the coin toss, the sample space of all outcomes S = {1,2}.

And with the dice, the sample space S = {1,2,3,4,5,6}

Now, let us toss a coin say 50 times, and record the outcome (heads or tails) each time. And let us say heads occurred 28 times, and tails occurred 22 times. According to this ‘experiment’ the probability of heads occurring is given as:

P (Heads) = , and the probability of tails occcuring is P (Tails) = .

This is called Experimental Probability – probability derived from a series of trials, or experiments. The probabilities obtained from experimental probability may differ as we increase the number of trials. With a very large number of trials or experiments, the probabilities may come close to the actual Theoretical Probability.

In theoretical probability, we expect the occurence of outcomes to be equally likely. When all outcomes are equally likely, then the theoretical probability of an event E is given by:

Probability of E or P(E)  = or P(E) = where n(E) = number of ways the event E can occur, and n(S) = total number of possible outcomes.

## Examples of probability of simple events:

Example 1: A committee has 8 female and 12 male members. What is the probability of choosing a female as the president of this committee?

Probability of choosing a female as a president = Here the number of females (favourable event) is 8, and the total number of members (outcomes) is 20. Hence

P (choosing a female president) = Example 2: The numbers 1 to 10 are written on separate pieces of paper, folded and put in a box. One number (piece of paper) is drawn from this box.

2a. What is the probability that this number chosen randomly is 3?

Sample space = {1,2,3,4,5,6,7,8,9,10}, so n(S) = 10.

There is only one 3 in the box, so n(E) = 1.

Hence P(3)  = 2b. What is the probability that this randomly chosen number is even?

Once again, n(S) = 10.

There are five even numbers in the box, so n(E)  =  5.

Hence P(even number)  = = 2c. What is the probability that this number chosen is a prime number?

As we saw earlier, n(S) = 10.

The prime numbers in the box are 2, 3, 5, and 7. So n(E)  =  4.

So P(prime number)  = = Example 3: A bag has 3 green, 2 red, 5 purple, 10 white and 5 black marbles.

3a. What is the probability of choosing a black ball?

Total marbles in the bag = 25.

Total black marbles = 5.

So P (black) = = 3b. What is the chance of picking a red or a green marble?

Total number of red and green marbles = 3 + 2  = 5

P (red or green) =  3c. What is the probability of choosing a brown marble?

There are no brown marbles in the bag.

So P (brown) = =  0

3d. What is the probability of choosing any coloured marble?

Since there are 25 coloured marbles in the bag, all the 25 marbles are favoured.

P (any coloured marble) = =  1.

We now look at probability of complementary events