Question

Two cards are randomly selected from a deck of 52 cards without replacement. Define two events:

A = { The first card is a King. }

B = { The second card is a King. }

1. 1. What is P (A)?
2. 2. What is P (B)?
3. 3. What is P (A ∩ B)?
   4. What is P (A|B)?
   5. What is P (A ∪ B)?
   6. Are A and B independent? Give your reason.
   7. Define a random variable Y be the number of Kings. Find the proba- bility distribution function of Y .
   8. What is the E(Y)?

Answer 1

n=52 cards.

A={the first card is a king}

B={the second card is a king}

without replacement.

total=52

total kings=4

(A)

P(A)==0.07692

P(A)=0.07692

(B)

P(B)==0.05882 (since its a case of without replacement so one king is taken total king left=3 and total cards=51)

P(B)=0.05882

(C)

P()=P(first card is king anf second card is king)==0.0045248

P()=0.004524

(D)

P()===0.07692

P()=0.07692

(E)

P()=P(A)+P(B)-P()

P()=0.07692+0.05882-0.004524

P()=0.13121

(F)

Two events A & B are independent if :

P()=P(A)*P(B)

now LHS=P()=0.004524

RHS=P(A)*P(B)=0.07692*0.05882=0.004524

since, LHS=RHS

hence A and B are independent.

(G)

Y= king is drawn.

total=52

kings=4

p(success)=

q(failure)=1-p=1-0.07692=0.92307

Y~binom(52,0.07692)

1.P(Y=0)=

2.P(Y=1)=

3.P(Y=2)=

4.P(Y=3)=

5,.P(Y=4)=

Y	0	1	2	3	4
Py(Y)	0.01556	0.06743	0.15348	0.19904	0.20315

(H)

E(Y)=

E(Y)=0*0.01556+1*0.06743+2*0.15348+3*0.19904+4*0.20315

E(Y)=0+0.06743+0.30696+0.59712+0.8126

E(Y)=1.78411

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