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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.
Questions and their Solutions
Question 1
A die is rolled, find the probability that an even number is obtained.
SolutionLet 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
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 i
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
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 numb
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
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 sho
there is one "3 of diamond" so that n(E) = 1 and n(S) = 52. Hence the probability of eve
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 t
are 4 "Queens" so that n(E) = 4 and n(S) = 52. Hence the probability of event E occurrin
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 draw
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 foll
color frequency
red 3
green 7
white 10
We now use the empirical formula of the probability
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
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 Ki
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 and links
elementary statistics and probabilities.
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