A standard deck of cards contains 52 cards. One card is selected from the deck.
(a) Compute the probability of randomly selecting a heart or club.
(b) Compute the probability of randomly selecting a heart or club or spade.
(c) Compute the probability of randomly selecting a four or spade.
Answer (2)
The probabilities of selecting cards from a standard deck are 0.5 for a heart or club, 0.75 for a heart or club or spade, and approximately 0.308 for a four or spade. Each probability was derived by considering the total number of specific card types within the 52-card deck. These calculations demonstrate how to apply basic probability principles in a practical scenario.
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Calculate the probability of selecting a heart or club: 52 13 + 52 13 = 0.5 .
Calculate the probability of selecting a heart or club or spade: 52 13 + 52 13 + 52 13 = 0.75 .
Calculate the probability of selecting a four or spade: 52 4 + 52 13 − 52 1 = 52 16 ≈ 0.308 .
The probabilities are: 0.5, 0.75, and 0.308, respectively. 0.5 , 0.75 , 0.308
Explanation
Analyze the problem We are given a problem involving probabilities of drawing cards from a standard 52-card deck. We need to calculate the probabilities for three different scenarios: drawing a heart or club, drawing a heart or club or spade, and drawing a four or spade.
Calculate P(heart or club) (a) To find the probability of drawing a heart or a club, we need to consider that there are 13 hearts and 13 clubs in a deck of 52 cards. Since a card cannot be both a heart and a club at the same time, these are mutually exclusive events. Therefore, we can simply add their probabilities: P ( h e a r t or c l u b ) = P ( h e a r t ) + P ( c l u b ) = 52 13 + 52 13 = 52 26 = 2 1 = 0.5 So, the probability of drawing a heart or a club is 0.5.
Calculate P(heart or club or spade) (b) To find the probability of drawing a heart or a club or a spade, we need to consider that there are 13 hearts, 13 clubs, and 13 spades in a deck of 52 cards. Since a card cannot be a heart, club, and spade at the same time, these are mutually exclusive events. Therefore, we can simply add their probabilities: P ( h e a r t or c l u b or s p a d e ) = P ( h e a r t ) + P ( c l u b ) + P ( s p a d e ) = 52 13 + 52 13 + 52 13 = 52 39 = 4 3 = 0.75 So, the probability of drawing a heart or a club or a spade is 0.75.
Calculate P(four or spade) (c) To find the probability of drawing a four or a spade, we need to consider that there are 4 fours and 13 spades in a deck of 52 cards. However, one of the fours is also a spade (the four of spades). So, we need to subtract the probability of drawing the four of spades to avoid double-counting: P ( f o u r or s p a d e ) = P ( f o u r ) + P ( s p a d e ) − P ( f o u r an d s p a d e ) = 52 4 + 52 13 − 52 1 = 52 16 = 13 4 ≈ 0.308 So, the probability of drawing a four or a spade is approximately 0.308.
State the final answer Therefore, the probabilities are: (a) Probability of selecting a heart or club: 0.5 (b) Probability of selecting a heart or club or spade: 0.75 (c) Probability of selecting a four or spade: 0.308
Examples
Understanding probabilities with card selections can be applied to various real-life scenarios. For instance, in games like poker or blackjack, knowing the likelihood of drawing certain cards can help players make informed decisions about betting and strategy. Similarly, in quality control, calculating the probability of selecting defective items from a production line can help assess the effectiveness of the manufacturing process and identify areas for improvement. These concepts are also fundamental in fields like finance, where understanding risk and probability is crucial for making investment decisions.
