Card deck Probability

Question

You pull out a card from a standard deck four times in a row, with replacement each time. Determine the probability that exactly two hearts turn up. 

Answer 1

The 52 deck of cards has 13 hearts, so the probability of drawing a heart would be 0.25, and the probability of not drawing a heart would be 0.75.

There are a couple ways to look at this problem.

A) Brute Force Method

We draw a sequence of four cards, and each card has two possible results: a heart or not a heart.

Thus, there are 2⁴ = 16 unique sequences we could draw.

We can figure out the probability of each sequence that has exactly two hearts, then add them up.

The 16 possible sequences (let H represent a heart and T represent not a heart):

HHHH

HHHT

HHTH

HTHH

THHH

HHTT

HTHT

HTTH

THHT

THTH

TTHH

HTTT

THTT

TTHT

TTTH

TTTT

We see that only six sequences have exactly two hearts: HHTT, HTHT, HTTH, THHT, THTH, TTHH.

The probability of HHTT is 0.25 * 0.25 * 0.75 * 0.75.

In fact, all six sequences that have exactly two hearts have the same probability.

Simply add them up to get your answer.

B) Binomial Distribution Method

1) Because we draw with replacement, every draw is independent from each-other.

2) Note that each draw event is a Bernoulli trial because it only has two possible outcomes. The draw could either turn up a heart or not turn up a heart.

3) By using the binomial distribution, we can find out the probability of getting exactly two hearts from drawing four times. Note that "If each Bernoulli trial is independent, then the number of successes in Bernoulli trails has a binomial Distribution." Here, we choose to define "number of successes" as "number of hearts".

4) Binomial Distribution Formula:

B = (_NC_X) * P^X * (1-P)^N-X.

Let N denote the number of trials. In our case, N=4 because we draw four times.

Let X denote the exact number of successes we want. In our case, X=2 because we want exactly two hearts.

Let P denote the probability of success for a single Bernoulli trial. In our case P = 0.25 because that is the probability of drawing a heart.

Let B denote the binomial probability, which is the probability of getting X successes from N trials. In our case, it is the probability of getting 2 hearts from 4 draws.

Note that _NC_X = n! / [x! * (n-x)!]

5) Plug in the numbers to get your answer!

C) Understanding the methods together.

If you look closely, you will see that the reason why the brute force method found six sequences is because ₄C₂ = 6. Also, the probability of each sequence was 0.25² * 0.75². In fact, the binomial distribution method simply multiplies the number of successful sequences by the probability of each sequence.

Answer 2

When discussing probabilities of events that do not affect each other (that is, independent events), if we want to find the probability that all of them occur, we multiply the probability that each one occurs. So you need the probability of drawing a heart and the probability of not drawing a heart and multiply two of the first times two of the second.

But then we also have to consider that you could draw them in any order. For example, you could draw the two hearts first: HHNN (where H=heart, N=not heart). Or you could draw HNHN, or any of several other patterns.

So start by finding the probability of drawing a heart and the probability of not drawing a heart and let us know what you get.

Answer 3

The probability of drawing a heart is 13/52 because there are 13 cards with hearts and 52 cards total in a standard deck.

Since this question says with replacement, you will select from 52 cards for each draw.

The probability of drawing a non-heart is 39/52 because 39 cards are not hearts. They are spades, clubs, and/or diamonds.

In order to draw exactly two hearts, we need the following to occur, in any order:

(13/52) * (13/52) * (39/52) * (39/52) = 0.0352

Answer 4

Since you are sampling with replacement, the probability that you will pick a heart (1/4) remains constant and picking a heart is a binomial variable and you can use the binomial distribution to find the probability of picking exactly 2 hearts:

C (4,2) * (1/4) ^2 * (3/4) ^2

Answer 5

13/52 * 13/52 * 39/52 * 39/52 =

(1/4)(1/4)(3/4)(3/4) = 9/256

Answer 6

The probability of drawing a heart (H) is 1/4. The probability of not drawing a heart (N) is 3/4.

There are 16 possible combinations of drawing either a heart (H) or no heart (N) in 4 draws. They are:

HHHH, HHHN, HHNH, HHNN, HNHH, HNHN, HNNH, HNNN, NHHH, NHHN, NHNH, NHNN, NNHH, NNHN, NNNH, NNNN

There are 6 of these possible combinations that provide exactly 2 hearts: HHNN, HNHN, HNNH, NHHN, NHNH, NNHH. Each of these has the same probability of being drawn: 1/4•1/4•3/4•3/4 = 9/256 but since there are 6 possibilities that provide that, we can multiply that by 6 to get 54/256 = 27/128

Answer 7

Let H = hearts, D = diamonds, C = clubs, and S = spades.

There are 4 suits. Therefore, we have (1 + 1 + 1 + 1)ⁿ = 4ⁿ.

Let n be the number of times a card is picked from the standard deck. Since there are 4 cards to pick out, then 4⁴ = 256 possible outcomes. These are the outcomes that only contains exactly 2 hearts.

HHCC HHCD HHCS

HHDC HHDD HHDS

HHSC HHSD HHSS

HCHC HCHD HCHS

HDHC HDHD HDHS

HSHC HSHD HSHS

HCCH HCDH HCSH

HDCH HDDH HDSH

HSCH HSDH HSSH

CHHC CHHD CHHS

DHHC DHHD DHHS

SHHC SHHD SHHS

CHCH CHDH CHSH

DHCH DHDH DHSH

SHCH SHDH SHSH

CCHH CDHH CSHH

DCHH DDHH DSHH

SCHH SDHH SSHH

P(exactly 2 hearts) = 54/256 = 27/128 = 0.2109375

Answer 8

With replacement or independence the probability of exactly 2 out of 4 events being events of probability p is

P = (Combination of 4 things 2 at a time) p²(1-p)²

The probability of a heart is 1/4 (p) and the combination of 4 things 2 at a time = 4*3/2 = 6

Take care.

Answer 9

A standard deck of playing cards (minus any Jokers) has 52 cards. Each card draw can be considered to be a single probabilistic event. Since the cards are replaced into the deck after being drawn, each event is independent from every other card draw event. In other words, each time you draw a card it's like the card is being drawn from a completely new deck of cards. Since the events are independent, we can multiply the probabilities of each card draw event together to get the probability of drawing certain combinations of cards. The probability of drawing a heart from the deck is 13/52, or 0.25. Therefore, the probability of not drawing a heart is 0.75. To get the probability of drawing exactly 2 hearts, we do 0.25 × 0.25 × 0.75 × 0.75, which gives us about 0.03516. (Which is equivalent to 9/256 or 3.516%.) Note that since each card draw event is independent of the other card draw events, the order in which the cards are drawn does not matter. We just need to get exactly 2 hearts. Mathematically this can be proven by the commutative property of multiplication. Hope this helps!