Final answer:
To find the probability that the third spade is drawn on the sixth try from a standard deck, we calculate the probability of drawing two spades and three non-spades in the first five cards and multiply that by the probability of drawing another spade from the remaining cards.
Step-by-step explanation:
The question involves calculating the probability of a specific event occurring when drawing cards from a standard deck without replacement. The deck consists of 52 cards with four suits (clubs, diamonds, hearts, spades), each suit having 13 cards. To find the probability that the third spade appears on the sixth draw without replacement, we will use combinatorial methods and conditional probability. We need to consider the following sequence of events: The first five cards contain exactly two spades and the sixth card drawn is a spade.
We calculate the number of ways to draw two spades from the thirteen available in the deck and three non-spades from the remaining 39 cards (not spades). There are C(13,2) ways to choose two spades and C(39,3) ways to choose three non-spades. The total number of ways to draw five cards from the entire deck is C(52,5). The probability of drawing two spades and three non-spades in the first five draws is
P(2S,3N) = (C(13,2) * C(39,3)) / C(52,5).
Given that two spades have already been drawn, there are 11 spades left out of 47 cards remaining in the deck. The probability of drawing the third spade on the sixth draw is 11/47. Therefore, the overall probability that the third spade appears on the sixth draw is given by multiplying the two probabilities:
P(3rd Spade on 6th Draw) = P(2S,3N) * (11/47)
This problem is an example of sampling without replacement and demonstrates the calculations of conditional probabilities in a combinatorial context.