Question

draws 13 cards from a deck of cards. what is the probability that at least 2 of cards are hearts

Answer 1

Answer:

Approximately 0.9

Step-by-step explanation:

P(at least 2 are hearts) = 1 - P(0 or 1 are hearts)

So let's work out P(0 hearts) and P(1 heart).

We'll need to use:

# ways of selecting 13 cards from 52 is $\displaystyle\binom{52}{13}$

Probability of getting 0 hearts

# ways of selecting 13 cards with none of them being a heart

= # ways of selecting 13 cards from the 52-13 = 39 non-heart cards

= $\displaystyle\binom{39}{13}$

So the probability of getting 0 hearts is

$\displaystyle P(\text{0 hearts}) = \frac{\dbinom{39}{13}}{\dbinom{52}{13}}$

Probability of getting exactly 1 heart

# ways of selecting 13 cards with exactly 1 of them being a heart

= ( # ways of selecting 1 heart from 13 ) × ( # ways of selecting 12 cards from the 39 non-heart cards )

= $\displaystyle13\binom{39}{12}$

So the probability of getting 1 heart is

$\displaystyle P(\text{1 heart})=\frac{13\dbinom{39}{12}}{\dbinom{52}{13}}$

Probability of getting at least 2 hearts

Now we just add those together to get P(0 or 1 hearts) and then subtract from 1 to get P(at least 2 hearts).

$\displaystyle P(\text{at least 2 hearts})=1 - \frac{\dbinom{39}{13}+13\dbinom{39}{12}}{\dbinom{52}{13}}=\frac{\dbinom{52}{13}-\dbinom{39}{13}-13\dbinom{39}{12}}{\dbinom{52}{13}}$

Numerical approximation

Using a computer, the values of the binomial coefficients here are:

$\displaystyle\binom{52}{13}=635013559600\\\\\binom{39}{13}=8122425444\\\\\binom{39}{12}=3910797436$

So the probability is:

$\displaystyle\frac{635013559600-8122425444-13\times3910797436}{635013559600}\\\\=\frac{576050767488}{635013559600}\\\\=\frac{43221096}{47645075}\\\\\approx0.90714719$