If you buy a lottery ticket in 50 lotteries, in

Kelly Nelson 2021-12-19 Answered
If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1100, what is the (approximate) probability that you will win a prize (a) at least once, (b) exactly once, and (c) at least twice?
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alexandrebaud43
Answered 2021-12-20 Author has 36 answers
Step 1
P{X=i}eλλii!
Given: n=50
p=1100=0.01
So, λ=np=50×0.01=0.5
a. Probability that you will win a prize at least once
P{X1}1P{X=0}=1e0.50.500!=0.3935
b. Probability that you will win a prize exactly once
P{X=0}e0.50.511!=0.3033
c. Probability that you will win a prize at least twice
P{X2}1P{X=0}P{X=1}=1e0.50.500!e0.50.511!=0.0902
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Jonathan Burroughs
Answered 2021-12-21 Author has 37 answers
Let X represent the number of winning prizes in 50 lotteries. The trials are Bernoulli trials.
Clearly, X has a binomial distribution with n=50 and p=1100
q=1p=11100=99100
(X=x)=nCxqnxpx=50Cx(99100)50x(1100)x
a) P (winning at least once) =P(X1)
=1P(X<1)
=1P(X=0)
=150Cx(99100)50
=11(99100)50
=1(99100)50
b) P (winning exactly once) =P(X=1)
=50C1(99100)49(1100)1
=50(1100)(99100)49
=12(99100)49
c) P (at least twice) =P(X2)
=1P(X<2)
=1P(X1)
=1[P(X=0)+P(X=1)]
=[1P(X=0)]P(X=1)
=1(99100)5012(99100)49
=1(99100)49[99100+12]
=1(99100)49(149100)
Answered 2021-12-27

Let X:Number of times he wins a prize
Winning a prize on lottery is a Bernoulli trial
So, X has a binomial distribution
P(X=x)=nCxqnxpx
Here, n=number of lotteries=50
p=Probability of winning a prize=1100
q=1p=11100=99100
HenceP(X=x)=50Cx(1100)x(99100)50x
a) Probability that he wins the lottery atleast once
P(at least once)=P(X1)
=1P(0)
=150C0(1100)0(99100)500
=11×1×(99100)50
=1(99100)50
b) Probability that he wins the lottery exactly once
P(exactly once)=P(X=1)
=50C1(1100)1(99100)501
=50×1100×(99100)49
=12(99100)49
c) P(at least twice)=P(X2)
=1[P(X=0)+P(X=1)]
=1[50C0(1100)0(99100)500+50C1(1100)1(99100)501]
=1[(99100)50+12(99100)49]
=1(99100)49[99100+12]
=1(99100)49[99+50100]
=1149100(99100)49

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