# Probability question?

In a 30 card deck

I have 4 A cards and I have 8 B cards

I draw 7 cards

Event A:

The cumulative probability of getting at least 1 A card is 0.68 according to http://stattrek.com/online-calculator/hypergeometr...

Event B:

The culmulative probability of getting at least 2 B cards is 0.62

The probability of event A and event B happening at the once is 0.62 x 0.68 = 0.42

Is the probability of event A and or event B

(0.62 + 0.68)/2 = 0.65?

Or should it be higher than 0.68%?

Should it be 0.62 + 0.68 - 0.42 = 0.88?

Thank you!

### 3 Answers

- MathMan TGLv 77 years agoFavourite answer
P( at least one A ) = 1 - P (no A's) =

1 - (26C7) / (30C7) = 0.67688 = 0.68

P( at least 2 B) =

1 - P(0 B) - P(1 B) =

1 - (22 choose 7)/(30 choose 7) - (8 * 22 choose 6) / 30 choose 7

= 0.70299 = 0.62302

~ ~ ~ Agree with your numbers so far ~ ~ ~

P(A and B) is not simply P(A) * P(B)

because they are not independent.

Intuitively, consider this:

P(B occurs) is based on having 7 chances to draw 2 B's.

Suppose the first card drawn is an A (and therefore event A has occurred).

Now you only have 6 shots at drawing 2 B's, which makes it less likely to occur.

That's over simplified, but it gives you the general idea:

If A occurs, B is somewhat less likely to occur,

and if B occurs, A is somewhat less likely to occur.

P( A and B) = 1 - (don't get an A and 2 B's)

This is a rather complex calculation.

If can fail as follows:

A B C

0 * * = 26C7

1+ 0 * = 22C7 - 18C7 [ all A-C only - C only ]

1+ 1 * = 8 * (22C6 - 18C6) [ 1 B and ( 6 A-C - C only ) ]

= 1244912

Total draws = 30 choose 7 = 2035800

P(not ( A & B) ) = 0.6115

P (A and B) = 1 - 0.6115 = 0.3885

Finally,

P( A or B) = P(A) + P(B) - P(A & B) = .68 + .62 - .39 = 0.91

~ ~ ~ end of disagreement section ~ ~ ~

And yes P(A or B) must be higher than either of them individually,

because A or B includes each of the other two as a subset.

Your formula P(A) + P(B) - P(A & B) is correct,

and produces the correct result once you have the right

value for P(A & B).

Source(s): All the above calculations verified by doing a simulation of 10000 deals and getting results that are very close to the numbers calculated. - DroningOnLv 67 years ago
it is 0.88 and yes, the probability of either event happening will always be more than both happening at same time

- Anonymous7 years ago
the answer is 42, that is all