probability, using direct and indirect reasoning?

Question

a student deals 3 cards from a statndard 52 deck of cards, what is the probability to the nearest hundredth at least one card is a heart. 

a) Direct reasoning

 

b) indirect reasoning

Answer 1

Hello, thank you for taking the time to post your question!

For the direct reasoning route I think that it’s just wanting you to sum across all of the scenarios where you could end up getting at least one card being a heart. That means finding P(exactly 1 heart and 2 non-hearts) + P(exactly 2 hearts and 1 non-heart) + P(exactly 3 hearts and 0 non-hearts). Using a calculator I find that value to be

(9633 + 3042 + 286) / 22100 = 0.59

For indirect reasoning I think it means you can use the idea of Complements … so like finding the probability of getting no hearts out of 3 cards and then subtracting that from 1.

That ends up being

P(no hearts) = (39 choose 3) / 22100 = 9139 / 22100 = 0.4135

P(at least one heart) = 1 – P(no hearts) = 1 – 0.4135 = 0.59

So you end up getting the same underlying probability value using either approach!

I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)

Answer 2

P(HNN) = 1/13 x 39/51 x 38/50

P(HHN) = 1/13 x 12/51 x39/50

P(HNH) = 1/13 x 39/51 x 12/50

P(HHH) = 1/13 x 12/51 x 11/50

P(NHN) = 39/52 x 13/51 x 38/51

P(NNH) = 39/52 x 38/51 x 13/50

P(NHH) = 39/52 x 13/51 x 12/50

sum up the above products = P(at least one heart in 3 draws)

or

just calculate the probability of zero hearts in 3 draws, then subtract that from 1

P(NNN) = 39/52 x 38/51 x 37/50

1- P(NNN) = 1 - 39/52 x 38/51 x 37/50 = 1 -54834/132600 = .586470588 = about 58.65% chance of at least one heart

one out of 4 cards is a heart

so, 1 -(3/4)^3 is a close estimate = about 57.81% which is close to 58.65%

57.81% is the probability if the 3 draws had been with replacement

3/4 x 3/4 x 3/4 = 27/64

1 -27/64 = (64-27)/64 = 37/64 = 0.5781