Question

four cards are drawn one by one with replacement from the well shuffled pack of 52 cards find the following probability of 1) exactly three spread 2)at least one club​

Answer 1

Answer:

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Step-by-step explanation:

Let X represent the number of spade cards among the five cards drawn. Since the drawing of card is with replacement, the trials are Bernoulli trials.

In a well shuffled deck of 52 cards, there are 13 spade cards.

⇒p=

52

13

​

=

4

1

​

∴q=1−

4

1

​

=

4

3

​

X has a binomial distribution with n=5 and p=

4

1

​

P(X=x)=

n

​

C

x

​

q

n−x

p

x

, where x=0,1,2,...n

=

5

​

C

x

​

(

4

3

​

)

5−x

(

4

1

​

)

x

(i) P(allfivecardsarespades)=P(X=5)

=

5

​

C

5

​

(

4

3

​

)

0

⋅(

4

1

​

)

5

=1⋅

1024

1

​

=

1024

1

​

(ii) P(only3cardsarespades)=P(X=3)

=

5

​

C

3

​

⋅(

4

3

​

)

2

⋅(

4

1

​

)

3

=10⋅

16

9

​

⋅

64

1

​

=

512

45

​

(iii) P(noneisaspade)=P(X=0)

=

5

​

C

0

​

⋅(

4

3

​

)

5

⋅(

4

1

​

)

0

=1⋅

1024

243

​

=

1024

243

​