To answer this question, we will compute the probability of not drawing a spade in a single draw and then use that calculation to find out the probability of not drawing a spade over three independent draws, with replacement.
Firstly, let's acknowledge that a standard deck of cards has 52 cards in total, with four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards. Therefore, there are 13 spades in a standard deck.
Now, the probability (P) of an event is the number of successful outcomes divided by the total number of possible outcomes. In the context of this problem, a successful outcome is drawing a card that is not a spade.
The number of non-spade cards in the deck is the total number of cards (52) minus the number of spade cards (13), which gives us 39 non-spade cards.
Now, let's calculate the probability of drawing a non-spade on one draw:
P(Non-Spade in one draw) = Number of Non-Spade Cards / Total Number of Cards
= 39 / 52
= 3 / 4 (simplifying the fraction by dividing both numerator and denominator by 13)
This is the probability for one draw. However, we need to find the probability of drawing three non-spade cards in a row with replacement. Because the draws are with replacement, the probability of each draw is independent and does not change from draw to draw.
So, to find the total probability of drawing three non-spade cards in three independent draws, we multiply the individual probabilities of each draw together:
P(Three Non-Spades) = P(Non-Spade in one draw) * P(Non-Spade in one draw) * P(Non-Spade in one draw)
= (3 / 4) * (3 / 4) * (3 / 4)
= 27 / 64 (multiplying the numerators together and denominators together)
Therefore, the probability that none of the chosen cards in three draws were spades, with replacement after each draw, is 27/64.
