Final answer:
The probability of drawing a heart remains approximately 1/4 upon each draw from a standard deck of playing cards when sampling without replacement because the ratio of hearts remaining to cards remaining stays close to 1/4, even as the number of cards in the deck decreases.
Step-by-step explanation:
The question involves calculating probabilities in the context of an ordinary deck of playing cards. Specifically, we are looking at the probability of drawing a heart at different stages of drawing cards without replacement. The deck of cards has 52 cards: 13 hearts, 13 diamonds, 13 spades, and 13 clubs. When a card is drawn and not replaced, this affects the total number of cards left in the deck but does not change the 1 in 4 chance of drawing a heart, assuming that we do not have the knowledge of the specific cards drawn before.
(a) The probability of drawing a heart as the first card is calculated as the number of hearts (13) divided by the total number of cards (52), which gives us p(First card is a heart) = 13/52 = 1/4.
(b) The probability of drawing a heart as the second card, assuming the first card was not a heart, remains 13 hearts out of 51 possible cards, p(Second card is a heart) = 13/51. However, the probability is still approximately 1/4 when rounded to two decimal places.
(c) Continuing without replacement, the probability of drawing a heart on the third draw is still roughly 1/4 for the same reasons, p(Third card is a heart) = 13/50.
(d) Similarly, the probability of drawing a heart on the fourth attempt again approximates to 1/4, p(Fourth card is a heart) = 13/49.
The calculations demonstrate that although the number of cards decreases with each draw, the ratio of hearts to the total number of cards remains close to 1/4, provided we do not know the specific cards that have been drawn. This approach can extend to additional draws with the ratio approximating to 1/4 at each draw.
When we do not replace cards, we are dealing with sampling without replacement, and each event is dependent on the previous ones. The probabilities change slightly with each draw, but if we don't know what cards have been previously drawn, the chance to draw a heart seems to be about 1/4 every time. The outcome of one draw doesn't affect the probability of the next draw of being a heart, considering we don't know the specific outcomes, hence maintaining the same chance approximately.