Statistics Question

Question

I need help with the following questions. I would really like to know how to solve/get the answer so I can understand. I have tried many times and I do not know what I am doing wrong.

If two cards are drawn from a standard 52 card playing deck, with replacement:

d. determine the probability of getting a face card and a number less than 5. Assume Ace is Low. Round to four decimal places. 

e. determine the probability of getting two hearts. Round to the nearest thousandth.

If two cards are drawn from a standard 52 card playing deck, without replacement:

f. determine the probability of getting a face card and a number less than 5. Assume Ace is Low. Round to four decimal places. 

g. determine the probability of getting two hearts. Round to the nearest thousandth. 

Answer 1

For question d)

I'm not sure if you are doing this or not, but pretend you are drawing one card, putting it back, and then drawing another card.

There are a total of 3 face cards per suit and 4 suits so 12 total face cards per deck. Regarding the numbers less than 5, I wouldn't normally count A (ace) as a "number" card because it does not have a number on it but their comment about "ace being low" infers that they want us to count ace as 1. Based on that, the numbers less than 5 are 1, 2, 3, and 4 (4 cards per suit) so there are 16 total cards less than 5.

So initially, let's think about drawing a face card then a card less than 5. For Draw #1, the probability of drawing a face card is 12/52 (equal to 3/13 by reducing the fraction). Then for Draw #2: the probability of drawing a card less than 5 is 16/52 (equal to 4/13 by reducing the fraction). The probability of drawing both (since they are independent of each other, meaning one draw does not affect the probability of the other draw) is 3/13 x 4/13 or 12/169.

Now, consider that you might draw the card less than 5 first, then the face card. The probability is still the same ( this time it will be 4/13 x 3/13 which is still 12/169). So the answer to question a) is 12/169 which is 0.0710 as a decimal with 4 places.

For question e)

There are 13 hearts so the probability of drawing a heart on Draw #1 is 13/52 (which can be reduced to 1/4) and the probability of drawing a heart on Draw #2 is also 1/4 (because you replaced the first card). The probability of both events happening is 1/4 x 1/4 which is 1/16 which is equal to 0.0625. But they ask for the answer to the nearest thousandth. Typically, when the last digit is 5, we round up, so the answer would be 0.063

For question f) (same as d but without replacement)

Assuming we will draw a face card then a number less than 5, the probability of drawing a face card on Draw #1 is the same (3/13) but the probability of drawing a number less than 5 is only 16/51 instead of 16/52. So the probability of both is 3/13 x 16/51 which equals 48/663 which can be reduced to 16/221 or equal to the decimal 0.0724

If you considered it the other way, first draw a number less than 5 then draw a face card, it still ends up the same.

For question g)

Draw #1: 13/52 = 1/4

Draw #2: Since you only care about drawing two hearts, assume you drew a heart on Draw #1 so now there are 12 hearts left and 51 total cards left so the probability for drawing a heart will be 12/51 (which can be reduced to 4/17). That means the probability of both events happening is 1/4 x 4/17 = 4/68 = 1/17 = 0.0588235 which is rounded to 0.059