Question #23921

two cards are drawn from a well shuffled pack of 52 cards and thrown away. find the probability of drawing an ace from the remaining 50 cards

Expert's answer

According to the total probability formula, we have

P(draw an ace from the remaining 50 cards) =

P(draw an ace | two drawn cards were aces) * P(draw two aces from 52 cards) +

P(draw an ace | two drawn cards were ace and not an ace) * P(draw one ace and one not ace from 52 cards) +

P(draw an ace | two drawn cards were not aces) * P(not to draw two aces from 52 cards).

Obviously,

P(draw an ace | two drawn cards were aces) = 0,

P(draw an ace | two drawn cards were ace and not an ace) = 3/50,

P(draw an ace | two drawn cards were not aces) = 4/50,

and

P(draw two aces from 52 cards) = 4/52 * 3/51,

P(draw one ace and one not ace from 52 cards) = 4/52 * (1 - 3/51) (or, equally, (1-4/52) * 4/51),

P(not to draw two aces from 52 cards) = (1 - 4/52)(1 - 4/51) (or, equally, 1 - 4/52 * 3/51).

So,

P(draw an ace from the remaining 50 cards) =

0 * 4/52 * 3/51 + 3/50 * 4/52 * (1 - 3/51) + 4/50 * (1 - 4/52)(1 - 4/51) =

= 3/50 * 4/52 * 48/51 + 4/50 * 48/52 * 47/51 = 9024/132600 = 1128/16575 ≈ 0.068.

P(draw an ace from the remaining 50 cards) =

P(draw an ace | two drawn cards were aces) * P(draw two aces from 52 cards) +

P(draw an ace | two drawn cards were ace and not an ace) * P(draw one ace and one not ace from 52 cards) +

P(draw an ace | two drawn cards were not aces) * P(not to draw two aces from 52 cards).

Obviously,

P(draw an ace | two drawn cards were aces) = 0,

P(draw an ace | two drawn cards were ace and not an ace) = 3/50,

P(draw an ace | two drawn cards were not aces) = 4/50,

and

P(draw two aces from 52 cards) = 4/52 * 3/51,

P(draw one ace and one not ace from 52 cards) = 4/52 * (1 - 3/51) (or, equally, (1-4/52) * 4/51),

P(not to draw two aces from 52 cards) = (1 - 4/52)(1 - 4/51) (or, equally, 1 - 4/52 * 3/51).

So,

P(draw an ace from the remaining 50 cards) =

0 * 4/52 * 3/51 + 3/50 * 4/52 * (1 - 3/51) + 4/50 * (1 - 4/52)(1 - 4/51) =

= 3/50 * 4/52 * 48/51 + 4/50 * 48/52 * 47/51 = 9024/132600 = 1128/16575 ≈ 0.068.

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