Tools of the trade
6.2 Probability distributions
The probability distribution of a discrete random variable is a display of all its possible values and their corresponding probabilities. The usual method of display is by tabulation in a probability distribution table. The probability distribution also can be represented in a vertical line graph or in a bar chart.
We learnt how to find probabilities for selections with and without replacement in Chapters 4 and 5, Sections 4.3, 4.4, 4.5 and 5.4.
REWIND
A variable is denoted by an upper-case letter and its possible values by the same lower-case letter.
If X can take values of 1, 2 and 3, we write X ∈ {1, 2, 3}, where the symbol ∈ means ‘is an element of’.
TIP
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Consider tossing two fair coins, where we can obtain 0, 1 or 2 heads.
The number of heads obtained in each trial, X, is a discrete random variable and X∈{0, 1, 2}. X
P( =0)=P(tails and tails)=0.5×0.5=0.25 X
P( = =1) P(heads and tails)+P(tails and heads)=(0.5×0.5) (0.5+ ×0.5)=0.5 X
P( =2)=P(heads and heads)=0.5×0.5=0.25
The probability distribution for X is displayed in the following table.
x 0 1 2
P(X==x) 0.25 0.5 0.25
The probabilities for the possible values of X are equal to the relative frequencies of the values. We would expect 25% of the tosses to produce zero heads; 50% to produce one head and 25% to produce two heads.
The following table shows the probability distribution for the random variable V.
v 2 3 4 5 6
P(V==v) 0.05 c2 c+0.1 2c+0.05 0.16 Find the value of the constant c and find P(V>4).
WORKED EXAMPLE 6.2
Note that ΣP(X=x) 1= . TIP
P(X=x) is equal to the relative frequency of each particular value of X.
TIP A fair square spinner with sides labelled 1, 2, 3 and 4 is spun twice. The two scores
obtained are added together to give the total, X. Draw up the probability distribution table for X.
Answer
x 2 3 4 5 6 7 8
X x
P( == ) 1
16 2 16
3 16
4 16
3 16
2 16
1
16 Sum=1
2nd spin
5 4 3 2 4 3 2 1
6 5 4 3
7 6 5 4
8 7 6 5
1 2 3 4
1st spin
WORKED EXAMPLE 6.1
The grid shows the 16 equally likely outcomes for the discrete random variable X, where X∈{2, 3, 4, 5, 6, 7, 8}.
The probability distribution for X is shown in the table.
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Answer
+ + + + + + =
+ =
− + =
c c c
c c
c c
0.05 0.1 2 0.05 0.16 1
3 – 0.64 0 ( 0.2)( 3.2) 0
2
2
c=0.2 or c= −3.2
The valid solution is c=0.2.
∴
P(V 4) P(V 5) P(V 6)(2 0.2) 0.05 0.16 0.61
> = = + =
= × + +
=
We use Σ =p 1 to form and solve an equation in c.
Note that if c= −3.2, then
V V
P( =3)=10.24, P( =4)= −3.1 and P(V=5)= −6.35.
Do check whether the solutions are valid.
Remember that a probability cannot be less than 0 or greater than 1.
TIP
There are spaces for three more passengers on a bus, but eight youths, one man and one woman wish to board.
The bus driver decides to select three of these people at random and allow them to board.
Draw up the probability distribution table for Y, the number of youths selected.
Answer
Selections are made without replacement, so we can use combinations to find P(Y =y). Possible values of Y are 1, 2 and 3
10C
3 possible selections.
Y C C
P( 1) C 1
15
8 1 2
2 10 3
= = × =
Y C C
P( 2) C 7
15
8 2 2
1 10 3
= = × =
Y C C
P( 3) C 7
15
8 3 2
0 10 3
= = × =
y 1 2 3
P(Y==y) 1 15
7 15
7
15 Always check that
1 Σ =p .
TIP WORKED EXAMPLE 6.3
Selecting three from 10 people.
At least one youth will be selected because there are only two non-youths, who we denote by Y′.
Selecting one from 8 , and two Y from 2Y′.
Selecting two from 8 , and one Y from 2Y′.
Selecting three from 8 , and none Y from 2Y′.
The table shows the probability distribution for Y.
A probability distribution shows all the possible values of a variable and the sum of the probabilities is
p 1 Σ =
KEY POINT 6.1
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EXERCISE 6A
1 The discrete random variable V is such that V∈{1, 2, 3}. Given that P(V = =1) P(V=2)= ×2 P(V =3), draw up the probability distribution table for V.
2 The probability distribution for the random variable X is given in the following table.
x 2 3 4 5
P(X==x) p 2p 1p
2 3p
Find the value of p and work out P(2<X <5).
3 The probability distribution for the random variable W is given in the following table.
w 3 6 9 12 15
P(W ==w) 2k k2 k
2 4
5−3k 13
50 a Form an equation using k, then solve it.
b Explain why only one of your solutions is valid.
c Find P(6øW<10) .
4 The probability that a boy succeeds with each basketball shot is 7
9. He takes two shots and the discrete random variable S represents the number of successful shots.
Show that P(S 0) 4
= =81 and draw up the probability distribution table for S.
5 At a garden centre, there is a display of roses: 25 are red, 20 are white, 15 are pink and 5 are orange.
Three roses are chosen at random.
a Show that the probability of selecting three red roses is approximately 0.0527.
b Draw up the probability distribution table for the number of red roses selected.
c Find the probability that at least one red rose is selected.
6 Three vehicles from a company’s six trucks, five vans, three cars and one motorbike are randomly selected and tested for roadworthiness.
a Show that the probability of selecting three vans is 2 91.
b Draw up the probability distribution table for the number of vans selected.
c Find the probability that, at most, one van is selected.
7 Five grapes are randomly selected without replacement from a bag containing one red grape and six green grapes.
Name and list the possible values of two discrete random variables in this situation.
State the relationship between the values of your two variables.
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8 A pack of five DVDs contains three movies and two documentaries. Three DVDs are selected and the following table shows the probability distribution for M, the number of movies selected.
m 1 2 3
P(M==m) 0.3 0.6 0.1
Draw up the probability distribution table for D, the number of documentaries selected.
9 In a particular country, 90% of the population is right-handed and 40% of the population has red hair.
Two people are randomly selected from the population. Draw up the probability distribution for X, the number of right-handed, red-haired people selected, and state what assumption must be made in order to do this.
10 A fair 4-sided die, numbered 1, 2, 3 and 5, is rolled twice. The random variable X is the sum of the two numbers on which the die comes to rest.
a Show that P(X 8) 1
= =8.
b Draw up the probability distribution table for X, and find P(X >6).
11 There are eight letters in a post box, and five of them are addressed to Mr Nut. Mr Nut removes four letters at random from the box.
a Find the probability that none of the selected letters are addressed to Mr Nut.
b Draw up the probability distribution table for N, the number of selected letters that are addressed to Mr Nut.
c Describe one significant feature of a vertical line graph or bar chart that could be used to represent the probability distribution for N.
12 A discrete random variable Y is such that Y∈{8, 9, 10}. Given that P(Y =y)=ky, find the value of the constant k.
13 Q is a discrete random variable and Q∈{3, 4, 5, 6}.
a Given that P(Q=q)=cq2, find the value of the constant c.
b Hence, find P(Q>4).
14 Four books are randomly selected from a box containing 10 novels, 10 reference books and 5 dictionaries.
The random variable N represents the number of novels selected.
a Find the value of P(N=2), correct to 3 significant figures.
b Without further calculation, state which of N=0 or N=4 is more likely. Explain the reasons for your answer.
15 In a game, a fair 4-sided spinner with edges labelled 0, 1, 2 and 3 is spun. If a player spins 1, 2 or 3, then that is their score. If a player scores 0, then they spin a fair triangular spinner with edges labelled 0, 1 and 2, and the number they spin is their score. Let the variable X represent a player’s score.
a Show that P(X 0) 1
= =12.
b Draw up the probability distribution table for X, and find the probability that X is a prime number.
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16 A biased coin is tossed three times. The probability distribution for H, the number of heads obtained, is shown in the following table.
h 0 1 2 3
P(H ==h) 0.512 0.384 0.096 a
a Find the probability of obtaining a head each time the coin is tossed.
b Give another discrete random variable that is related to these trials, and calculate the probability that its value is greater than the value of H.
17 Two ordinary fair dice are rolled. A score of 3 points is awarded if exactly one die shows an odd number and there is also a difference of 1 between the two numbers obtained. A player who rolls two even numbers is awarded a score of 2 points, otherwise a player scores 1 point.
a Draw up the probability distribution table for S, the number of points awarded.
b Find the probability that a player scores 3 points, given that the sum of the numbers on their two dice is greater than 9.
18 The discrete random variable R is such that R∈{1, 3, 5, 7}. a Given that R r k r
= = r +
P( ) (+ 1)
2 , find the value of the constant k.
b Hence, find P(Rø4).
We will learn how to extend this Explore activity to more than two coins in Chapter 7.
We will see how to represent the probabil- ity distribution for a continuous random variable in Chapter 8, Section 8.1.
FAST FORWARD EXPLORE 6.1
Consider the probability distribution for X, the number of heads obtained when two fair coins are tossed, which was given in the table presented in the introduction of Section 6.2. Sketch or simply describe the shape of a bar chart (or vertical line graph) that can be used to represent this distribution.
In this activity, you will investigate how the shape of the distribution of X is altered when two unfair coins are tossed; that is, when the probability of obtaining heads is p≠0.5.
Consider the case in which p=0.4 for both coins. Draw a bar chart to represent the probability distribution of X, the number of heads obtained.
Next consider the case in which p=0.6 for both coins, and draw a bar chart to represent the probability distribution of X.
What do you notice about the bar charts for p=0.4 and p=0.6?
Investigate other pairs of probability distributions for which the values of p add up to 1, such as p=0.3 and p=0.7. Make general comments to summarise your results.
Investigate how the value of P(X =1) changes as p increases from 0 to 1, and then represent this graphically. On the same diagram, show how the values of P(X =0) and P(X =2) change as p increases from 0 to 1.
M
This can be done manually or using the Coin Flip Simulation on the GeoGebra website.
WEB LINK
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0.4 P(X = x)
0.2
0 1 2 3 x
0