3 we draw 6 cards from a normal 52-card deck, puting the cards Back into the dec

Question

3 we draw 6 cards from a normal 52-card deck, puting the cards Back into the deck and shuffling each time: (write the formula or calculator function you used to get each answer) a) what is P(exactly 3 hearts)? b) what is P(less than 3 hearts)? c) what is P(more than 3 hearts)? d) what is P(Hearts on the first two draws) e) what is P(Hearts on the first two draws, and not-Hearts on the last 4 draws) f) Use the results from d) and e) to compute P(No hearts on the last 4 draws | Hearts on the first two draws).

Explanation / Answer

Solution:

We are given n = 6

Sampling used is simple random sampling with replacement.

So, we can use binomial distribution.

Part a

Total number of hearts = 4

Total number of cards = 52

Probability for hearts = p = 4/52 = 0.076923

We have to find P(X=3) for B(n = 6, p = 0.076923)

P(X=x) = nCx*p^x*(1 – p)^(n – x)

P(X=3) = 6C3* 0.076923^3*(1 - 0.076923)^(6 – 3)

P(X=3) = 0.00716

Required probability = 0.00716

Part b

We have to find P(X<3)

P(X<3) = P(X=0) + P(X=1) + P(X=2)

We have

n = 6, p = 0.076923

P(X=0) = 6C0* 0.076923^0*(1 - 0.076923)^(6 – 0) = 0.618625

P(X=1) = 6C1* 0.076923^1*(1 - 0.076923)^(6 – 1) = 0.309312

P(X=2) = 6C2* 0.076923^2*(1 - 0.076923)^(6 – 2) = 0.06444

P(X<3) = P(X=0) + P(X=1) + P(X=2)

P(X<3) = 0.618625 + 0.309312 + 0.06444

P(X<3) = 0.992377

Required probability = 0.992377

Part c

Here, we have to find P(X>3)

P(X>3) = 1 – P(X3)

P(X3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)

P(X=0) = 6C0* 0.076923^0*(1 - 0.076923)^(6 – 0) = 0.618625

P(X=1) = 6C1* 0.076923^1*(1 - 0.076923)^(6 – 1) = 0.309312

P(X=2) = 6C2* 0.076923^2*(1 - 0.076923)^(6 – 2) = 0.06444

P(X=3) = 6C3* 0.076923^3*(1 - 0.076923)^(6 – 3) = 0.00716

P(X3) = 0.618625 + 0.309312 + 0.06444 + 0.00716 = 0.999537

P(X>3) = 1 – P(X3) = 1 - 0.999537 = 0.000463

Required probability = 0.000463

Part d

We have to find P(hearts on first two draws)

P(hearts on first two draws) = P(heart on first draw) * P(heart on second draw

P(hearts on first two draws) = 0.076923*0.076923 = 0.005917

Required probability = 0.005917

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