Question; Determine the probability of...

Question

Determine the probability of choosing a heart or a ten in a deck of cards. Show all steps please, thank you.

Answer 1

Well in a deck of cards there are 52 cards

There are four tens, and twelve hearts (that are not tens)

So 4 + 12 = 16 cards that are hearts or tens

So probability = 16 / 52 = 8 / 26 = 4 / 13.

Answer 2

In order to work out a probability with the word "or" in it, we add the two probabilities together, but there is a catch that this exact problem is built to test, so let us go slowly.

Start with what a standard deck looks like. There are 52 cards, split into 4 suits of 13 cards each, and there are 4 of every rank, one in each suit.

So the probability of drawing a heart is:

P(heart) = 13/52

And the probability of drawing a ten is:

P(ten) = 4/52

Now, if we just add those we get 17/52, but that answer is wrong, and here is why. One single card, the ten of hearts, is a heart AND a ten. When we counted the 13 hearts we already counted it, and when we counted the 4 tens we counted it a second time. So it got counted twice, and we need to take it back out once.

That is what the addition rule for probability is really saying:

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

The overlap here is just the single ten of hearts, so P(heart and ten) = 1/52. Putting it together:

P(heart or ten) = 13/52 + 4/52 - 1/52 = 16/52

Now we simplify. Both 16 and 52 divide by 4:

16/52 = 4/13

You can also check this by counting directly. There are 13 hearts, plus the ten of clubs, the ten of spades and the ten of diamonds, which is 16 cards out of 52.

So, our final answer is 4/13, or about 0.308.