Simple Probability

The probability of an event E happening, P(E), is given by the number of ways the event can occur, n(E), compared with the total number of outcomes possible n(S) (the size of the sample space).

 

P(E)=\cfrac { n(E) }{ n(S) }

 

If P(E) = 0 the event is impossible, and if P(E) = 1 the event is certain.

 

0 \le P(E) \le 1

 

The sum of all (mutually exclusive) probabilities is 1.

 

Examples

 

1. A container holds 8 blue, 7 white and 10 yellow marbles. If one marble is selected at random, find the probability of getting

a) a white marble

b) a white or blue marble

c) a yellow, white or blue marble

d) a red marble

Show Answer

The size of the sample space, or total number of marbles is 8 + 7 + 10 = 25.

a) P(W)=\cfrac{7}{15}

 

b) P(W or B)=\cfrac{8+7}{25}=\cfrac{15}{25}=\cfrac{3}{5}

 

c) P(W or B)=\cfrac{8+7+10}{25}=\cfrac{25}{25}=1

 

d) \cfrac{0}{15} chance - impossible.

 

2. The probability that a traffic light will turn green as a car approaches it is estimated to be \cfrac{5}{12}. A taxi goes through 288 intersections where there are traffic lights. How many of these would be expected to turn green as the taxi approached?

Show Answer

As the expected chance is only \cfrac{5}{12}, then the theoretical solution is:

\cfrac{5}{12} \times 288 = 120 times.

 

Complementary Events

 

The complement of an event happening is the event not happening. That is , the complement of P(E) is P(not E). We can write this as P\widetilde { (E) } .

For example:

A die is thrown. Find the probability of:

a) throwing a 4.

b) not throwing a 4.

Show Answer

a) P(4)=\cfrac{1}{6}

b) P(not 4)=P(1, 2, 3, 5, 6)=\cfrac{5}{6}

 

In general,

P(E)=1-P\widetilde { (E) }

 

Non-Mutually Exclusive Events

 

Sometimes, there is an overlap where more than one event can occur at the same time. We call these non-mutually exclusive events. It is important to count the possible outcomes carefully when this happens - and Venn diagrams help.

For example:

One card is drawn from a set of cards numbered 1 to 12. Find the probability of drawing out an odd number OR a multiple of 3.

The odd cards are 1, 3, 5, 7, 9, 11.

The multiples of 3 are 3, 6, 9 and 12.

The numbers 3 and 9 are both odd and multiples of 3 (- try not to count them twice).

So there are 8 numbers that are odd OR multiples of 3: 1, 3, 5, 6, 7, 9, 11, 12.

P(odd or multiples of 3) = \cfrac{6}{10}=\cfrac{3}{5}

We can also display this in a Venn Diagram:

 

In fact, there is a formula that can be used for non-mutually exclusive events.

P(A or B)=P(A)+P(B)-P(A and B)

 

From 50 cards numbered from 1 to 50, one is selected at random. Find the probability that the card selected is even or less than 15.

Some cards are both even and less than 15 = 2, 4, 6, 8, 10, 12, 14

P(even and < 15) = \cfrac{7}{50}

P(even) = \cfrac{25}{50}

P(< 15) = \cfrac{14}{50}

 

P(even or < 15) = P(even) + P(< 15) - P(even and < 15)

P(even or < 15) = \cfrac{25}{50} + \cfrac{14}{50} - \cfrac{7}{50}

P(even or < 15) = \cfrac{32}{50} = \cfrac{16}{25}