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

Relevance
  • 6 years ago
    Best 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.
  • 6 years ago

    it is 0.88 and yes, the probability of either event happening will always be more than both happening at same time

  • Anonymous
    6 years ago

    the answer is 42, that is all

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