Tutorial on finding the probability of an event.

In what follows, S is the sample space of the experiment in question and E is the event of interest. n(S) is the number of elements in the sample space S and n(E) is the number of elements in the event E.

__Question 1:__ A die is rolled, find the probability that an even number is obtained.

** Solution**
Let us first write the sample space S of the experiment.

S = {1,2,3,4,5,6}

Let E be the event "an even number is obtained" and write it down.

E = {2,4,6}

We now use the formula of the classical probability.

P(E) = n(E) / n(S) = 3 / 6 = 1 / 2

__Question 2:__ Two coins are tossed, find the probability that two heads are obtained.

**Note:** Each coin has two possible outcomes H (heads) and T (Tails).

__Solution__

The sample space S is given by.

S = {(H,T),(H,H),(T,H),(T,T)}

Let E be the event "two heads are obtained".

E = {(H,H)}

We use the formula of the classical probability.

P(E) = n(E) / n(S) = 1 / 4

__Question 3:__ Which of these numbers cannot be a probability?

a) -0.00001

b) 0.5

c) 1.001

d) 0

e) 1

f) 20%

__Solution__

A probability is always greater than or equal to 0 and less than or equal to 1, hence only **a)** and **c)** above cannot represent probabilities: -0.00010 is less than 0 and 1.001 is greater than 1.

__Question 4:__ Two dice are rolled, find the probability that the sum is

a) equal to 1

b) equal to 4

c) less than 13

__Solution__

a) The sample space S of two dice is shown below.

S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)

(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }

Let E be the event "sum equal to 1". There are no outcomes which correspond to a sum equal to 1, hence

P(E) = n(E) / n(S) = 0 / 36 = 0

b) Three possible outcomes give a sum equal to 4: E = {(1,3),(2,2),(3,1)}, hence.

P(E) = n(E) / n(S) = 3 / 36 = 1 / 12

c) All possible outcomes, E = S, give a sum less than 13, hence.

P(E) = n(E) / n(S) = 36 / 36 = 1

__Question 5:__ A die is rolled and a coin is tossed, find the probability that the die shows an odd number and the coin shows a head.

__Solution__

The sample space S of the experiment described in question 5 is as follows

S = { (1,H),(2,H),(3,H),(4,H),(5,H),(6,H)

(1,T),(2,T),(3,T),(4,T),(5,T),(6,T)}

Let E be the event "the die shows an odd number and the coin shows a head". Event E may be described as follows

E={(1,H),(3,H),(5,H)}

The probability P(E) is given by

P(E) = n(E) / n(S) = 3 / 12 = 1 / 4

__Question 6:__ A card is drawn at random from a deck of cards. Find the probability of getting the 3 of diamond.

__Solution__

The sample space S of the experiment in question 6 is shwon below

Let E be the event "getting the 3 of diamond". An examination of the sample space shows that there is one "3 of diamond" so that n(E) = 1 and n(S) = 52. Hence the probability of event E occurring is given by

P(E) = 1 / 52

__Question 7:__ A card is drawn at random from a deck of cards. Find the probability of getting a queen.

__Solution__

The sample space S of the experiment in question 7 is shwon above (see question 6)

Let E be the event "getting a Queen". An examination of the sample space shows that there are 4 "Queens" so that n(E) = 4 and n(S) = 52. Hence the probability of event E occurring is given by

P(E) = 4 / 52 = 1 / 13

__Question 8:__ A jar contains 3 red marbles, 7 green marbles and 10 white marbles. If a marble is drawn from the jar at random, what is the probability that this marble is white?

__Solution__

We first construct a table of frequencies that gives the marbles color distributions as follows

color | frequency |
---|---|

red | 3 |

green | 7 |

white | 10 |

P(E) = Frequency for white color / Total frequencies in the above table

= 10 / 20 = 1 / 2

__Question 9:__ The blood groups of 200 people is distributed as follows: 50 have type **A** blood, 65 have **B** blood type, 70 have **O** blood type and 15 have type **AB** blood. If a person from this group is selected at random, what is the probability that this person has O blood type?

__Solution__

We construct a table of frequencies for the the blood groups as follows

group | frequency |
---|---|

a | 50 |

B | 65 |

O | 70 |

AB | 15 |

We use the empirical formula of the probability

P(E) = Frequency for O blood / Total frequencies

= 70 / 200 = 0.35

__Exercises__

a) A die is rolled, find the probability that the number obtained is greater than 4.

b) Two coins are tossed, find the probability that one __head__ only is obtained.

c) Two dice are rolled, find the probability that the sum is equal to 5.

d) A card is drawn at random from a deck of cards. Find the probability of getting the King of heart.

__Answers to above exercises:__

a) 2 / 6 = 1 / 3

b) 2 / 4 = 1 / 2

c) 4 / 36 = 1 / 9

d) 1 / 52

More references on elementary statistics and probabilities.

Updated: 30 July 2018 (A Dendane)