a person purchase a lottery ticket in he wins chances of winning is time is 1 by hundred find the number of time he has to purchase a ticket to have 50% chance of winning
Answers
Step-by-step explanation:
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=
100
1
∴q=1−p=1−
100
1
=
100
99
∴P(X=x)=
n
C
x
q
n−x
p
x
=
50
C
x
(
100
99
)
50−x
⋅(
100
1
)
x
(a) P(winningatleastonce)=P(X≥1)
=1−P(X<1)
=1−P(X=0)
=1−
50
C
x
(
100
99
)
50
=1−1⋅(
100
99
)
50
=1−(
100
99
)
50
(b) P(winningexactlyonce)=P(X=1)
=
50
C
1
(
100
99
)
49
⋅(
100
1
)
1
=50(
100
1
)(
100
99
)
49
=
2
1
(
100
99
)
49
(c) P(atleasttwice)=P(X≥2)
=1−P(X<2)
=1−P(X≤1)
=1−[P(X=0)+P(X=1)]
=[1−P(X=0)]−P(X=1)
=1−(
100
99
)
50
−
2
1
⋅(
100
99
)
49
=1−(
100
99
)
49
[
100
99
+
2
1
]
=1−(
100
99
)
49
⋅(
100
149
)
=1−(
100
149
)(
100
99
)
49
Ex 13.5, 10
Chapter 13 Probability Class 12
