Expectation and variance of the binomial distribution
7.2 The geometric distribution
Consider a situation in which we are attempting to roll a 6 with an ordinary fair die.
How likely are we to get our first 6 on the first roll; on the second roll; on the third roll, and so on?
We can answer these questions using the constant probabilities of success and failure:
p and 1 – p.
P(first 6 on first roll)= →p a success.
P(first 6 on second roll)=(1 – ) p p→a failure followed by a success.
P(first 6 on third roll)=(1 – )p2p→ two failures followed by a success.
The distribution of X, the number of trials up to and including the first success in a series of repeated independent trials, is a discrete random variable whose distribution is called a geometric distribution.
The following table shows the probability that the first success occurs on the rth trial.
r 1 2 3 4 .... n ....
X r
P( == ) p p(1 – )p p(1 – )p2 p(1 – )p3 .... p(1 – )pn−1 ....
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The values of P(X =r) in the previous table are the terms of a geometric progression (GP) with first term p and common ratio 1 – . The sum of the probabilities is equal to the sum p to infinity of the GP.
[P( )] first term
1 common ratio 1 (1 ) 1
X r S p
∑ = = = p
− =
− − =
∞ .
The sum of the probabilities in a geometric probability distribution is equal to 1.
A discrete random variable, X, is said to have a geometric distribution, and is defined by its parameter p, if it meets the following criteria.
•
The repeated trials are independent.•
The repeated trials can be infinite in number.•
There are just two possible outcomes for each trial (i.e. success or failure).•
The probability of success in each trial, p, is constant.We saw in Chapter 6, Section 6.2 that Σ =p 1 for a probability distribution. You will also have seen geometric progressions and geometric series in Pure Mathematics 1, Chapter 6.
REWIND
An alternative form of this formula,
X=r =qr ×p
P( ) –1 ,
where p = 1 − q, reminds us that the
– 1
r failures occur before the first success.
TIP
Recall from Section 7.1 that
n
r C n
r n r
n r
!
!( )!
= =
− .
REWIND The binomial and geometric distributions arise in very similar situations. The significant
difference is that the number of trials in a binomial distribution is fixed from the start and the number of successes are counted, whereas, in a geometric distribution, trials are repeated as many times as necessary until the first success occurs.
For X~ B( , ), there are n p
n
r ways to obtain r successes.
For X~ Geo( ), there is only one way to obtain the first success on the rth trial, and that p is when there are r– 1 failures followed by a success.
Repeated independent trials are carried out in which the probability of success in each trial is 0.66.
Correct to 3 significant figures, find the probability that the first success occurs:
a on the third trial
b on or before the second trial c after the third trial.
WORKED EXAMPLE 7.7
A random variable X that has a geometric distribution is denoted by X~ Geo( )p, and the probability that the first success occurs on the rth trial is
(1 – ) –1
=
pr p pr for r=1, 2, 3,… KEY POINT 7.3
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Answer
a P( 3) (1 – ) 0.66 0.34 0.0763
2 2
= =
= ×
=
X p p
b X X X
p p p
= = + =
= +
=
P( ø2) P( 1) P( 2) (1 – )
0.884
c X X
X X X
p p p p p
> =
= = + = + =
= + +
=
P( 3) 1 – P( ø3)
1 – [P( 1) P( 2) P( 3)]
1 – [ (1 – ) (1 – ) ] 0.0393
2
Let X represent the number of trials up to and including the first success, thenX ~ Geo(0.66), where p=0.66 and 1 –p=0.34.
Probabilities that involve inequalities can be found by summation for small values of r, as in parts b and c of Worked example 7.7. However, for larger values of r, the following results will be useful.
Xør =
P( ) P(success on one of the first r trials)=1 – P(failure on the first r trials) X>r =
P( ) P(first success after the rth trial)=P(failure on the first r trials) These two results are written in terms of q in Key point 7.4.
In a particular country, 18% of adults wear contact lenses. Adults are randomly selected and interviewed one at a time. Find the probability that the first adult who wears contact lenses is:
a one of the first 15 interviewed b not one of the first nine interviewed.
Answer
a =
= −
= P( ø15) 1 –
1 0.82 0.949
15 15
X q
b P( 9) 0.82 0.168
9 9
> =
=
=
X q
WORKED EXAMPLE 7.8
Let X represent the number of adults interviewed up to and including the first one who wears contact lenses, then X ~ Geo(0.18) and q=1 – 0.18=0.82.
When X~ Geo( )p and q=1 –p, then
• P(X<r) 1 –= qr
• P(X> =r) qr KEY POINT 7.4
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A coin is biased such that the probability of obtaining heads with each toss is equal to 5
11. The coin is tossed until the first head is obtained. Find the probability that the coin is tossed:
a at least six times b fewer than eight times.
Answer
a X X
q
= >
=
=
=
P( ù6) P( 5) 6 11 0.0483
5 5
b X X
q
< =
= −
= −
=
P( 8) P( ø7) 1
1 6
11 0.986
7 7
WORKED EXAMPLE 7.9
Let X represent the number of times the coin is tossed up to and including the first heads, then
~ Geo 5 11
X and
= q 6
11.
EXERCISE 7C
1 Given the discrete random variable X ~ Geo(0.2), find:
a P(X =7) b P(X≠5) c P(X>4) . 2 Given that T~ Geo(0.32), find:
a P(T=3) b P(Tø6) c P(T>7) .
3 The probability that Mike is shown a yellow card in any football match that he plays is 1
2. Find the probability that Mike is next shown a yellow card:
a in the third match that he plays b before the fourth match that he plays.
4 On average, Diya concedes one penalty in every six hockey matches that she plays. Find the probability that Diya next concedes a penalty:
a in the eighth match that she plays b after the fourth match that she plays.
5 The sides of a fair 5-sided spinner are marked 1, 1, 2, 3 and 4. It is spun until the first score of 1 is obtained.
Find the probability that it is spun:
a exactly twice b at most five times c at least eight times.
‘At least six times’ has the same meaning as
‘more than five times’.
TIP
‘Fewer than eight times’ has the same meaning as ‘seven or fewer times’.
TIP
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6 It is known that 80% of the customers at a DIY store own a discount card. Customers queuing at a checkout are asked if they own a discount card.
a Find the probability that the first customer who owns a discount card is:
i the third customer asked ii not one of the first four customers asked.
b Given that 10% of the customers with discount cards forget to bring them to the store, find the probability that the first customer who owns a discount card and remembered to bring it to the store is the second customer asked.
7 In a manufacturing process, the probability that an item is faulty is 0.07. Items from those produced are selected at random and tested.
a Find the probability that the first faulty item is:
i the 12th item tested ii not one of the first 10 items tested iii one of the first eight items tested.
b What assumptions have you made about the occurrence of faults in the items so that you can calculate the probabilities in part a?
8 Two independent random variables are X ~ Geo(0.3) and Y ~ Geo(0.7). Find:
a P(X =2) b P(Y =2) c P(X =1 andY =1) .
9 On average, 14% of the vehicles being driven along a stretch of road are heavy goods vehicles (HGVs). A girl stands on a footbridge above the road and counts the number of vehicles, up to and including the first HGV that passes. Find the probability that she counts:
a at most three vehicles b at least five vehicles.
10 The probability that a woman can connect to her home Wi-Fi at each attempt is 0.44. Find the probability that she fails to connect until her fifth attempt.
11 Decide whether or not it would be appropriate to model the distribution of X by a geometric distribution in the following situations. In those cases for which it is not appropriate, give a reason.
a A bag contains two red sweets and many more green sweets. A child selects a sweet at random and eats it, selects another and eats it, and so on. X is the number of sweets selected and eaten, up to and including the first red sweet.
b A monkey sits in front of a laptop with a blank word processing document on its screen. X is the number of keys pressed by the monkey, up to and including the first key pressed that completes a row of three letters that form a meaningful three-letter word.
c X is the number of times that a grain of rice is dropped from a height of 2 metres onto a chessboard, up to and including the first time that it comes to rest on a white square.
d X is the number of races in which an athlete competes during a year, up to and including the first race that he wins.
12 The random variable T has a geometric distribution and it is given that =
= =
T T
P( 2)
P( 5) 15.625. Find P(T =3).
13 X ~ Geo( ) and p P(X =2)=0.2464. Given that p<0.5, find P(X >3).
14 Given that X ~ Geo( ) and that p P(Xø4) 2385
= 2401, find P(1øX <4). PS
PS
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15 Two ordinary fair dice are rolled simultaneously. Find the probability of obtaining:
a the first double on the fourth roll
b the first pair of numbers with a sum of more than 10 before the 10th roll.
16 X ~ Geo(0.24) and Y ~ Geo(0.25) are two independent random variables. Find the probability that X Y+ =4.
PS
PS