Giulia is playing a game with 6 cards: 4 kings, 1 queen, and 1 jack. She draws 1 card out of the stack of cards, replaces it, and then draws another card.
What is the probability that she will draw a king and then a jack, [tex]P( king, then jack )[/tex] ?
[tex] \frac{1}{36} [/tex]
[tex] \frac{1}{24} [/tex]
[tex] \frac{1}{9} [/tex]
[tex] \frac{1}{6} [/tex]
Answer (1)
Calculate the probability of drawing a king: 6 4 = 3 2 .
Calculate the probability of drawing a jack: 6 1 .
Multiply the probabilities since the events are independent: 3 2 × 6 1 = 9 1 .
The probability of drawing a king and then a jack is 9 1 .
Explanation
Understand the problem We are given that Giulia is playing a game with 6 cards: 4 kings, 1 queen, and 1 jack. She draws a card, replaces it, and then draws another card. We want to find the probability that she will draw a king and then a jack.
Calculate the probability of drawing a king The probability of drawing a king on the first draw is the number of kings divided by the total number of cards. There are 4 kings and 6 total cards, so the probability of drawing a king is 6 4 = 3 2 .
Calculate the probability of drawing a jack Since Giulia replaces the card, the second draw is independent of the first draw. The probability of drawing a jack on the second draw is the number of jacks divided by the total number of cards. There is 1 jack and 6 total cards, so the probability of drawing a jack is 6 1 .
Calculate the probability of drawing a king and then a jack Since the two draws are independent, we can multiply the probabilities to find the probability of drawing a king and then a jack: P ( king, then jack ) = P ( king ) × P ( jack ) = 3 2 × 6 1 = 18 2 = 9 1 .
State the final answer Therefore, the probability that Giulia will draw a king and then a jack is 9 1 .
Examples
This type of probability calculation is useful in many games of chance, such as card games or lotteries, where you need to determine the likelihood of specific sequences of events occurring. For example, if you were playing a card game and needed to draw a specific card to win, you could use this method to calculate your chances of drawing that card in a particular order. Understanding these probabilities can help you make more informed decisions and strategize effectively in such games. It also forms the basis for understanding more complex probability scenarios in fields like finance and data analysis.
