A board game uses the deck of 20 cards shown to the right. Two cards are selected at random from this deck. Calculate the probability that the first card selected has a yellow bird and the second card selected has a frog, both with and without replacement.
Two cards are to be selected with replacement. Determine the probability that the first card selected has a yellow bird and the second card selected has a frog.
Answer (2)
Calculate the probability of selecting a yellow bird: 20 4 = 5 1 .
Calculate the probability of selecting a frog: 20 3 .
Multiply the probabilities since the events are independent: 5 1 × 20 3 = 100 3 .
The probability of selecting a yellow bird and then a frog with replacement is: 100 3 .
Explanation
Understand the problem We are given a deck of 20 cards. We want to find the probability of selecting a card with a yellow bird first, and then a card with a frog second, with replacement. This means after we pick the first card, we put it back into the deck before picking the second card.
Calculate the probability of selecting a yellow bird First, let's find the probability of selecting a card with a yellow bird on the first draw. There are 4 cards with a yellow bird out of a total of 20 cards. So, the probability is: P ( ye ll o w bi r d ) = 20 4 We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4: P ( ye ll o w bi r d ) = 20 ÷ 4 4 ÷ 4 = 5 1
Calculate the probability of selecting a frog Next, let's find the probability of selecting a card with a frog on the second draw. Since we are drawing with replacement, the total number of cards remains 20. There are 3 cards with a frog. So, the probability is: P ( f ro g ) = 20 3
Calculate the probability of both events Now, we need to find the probability of both events happening. Since the two events are independent (because we replaced the first card), we can multiply their probabilities: P ( ye ll o w bi r d an d f ro g ) = P ( ye ll o w bi r d ) × P ( f ro g ) = 5 1 × 20 3 Multiplying the fractions, we get: P ( ye ll o w bi r d an d f ro g ) = 5 × 20 1 × 3 = 100 3
State the final answer Therefore, the probability that the first card selected has a yellow bird and the second card selected has a frog, with replacement, is 100 3 .
Examples
Imagine you're designing a simple card game where players draw two cards in sequence. This probability calculation helps you understand the likelihood of specific card combinations appearing, which is crucial for balancing the game's mechanics and ensuring fair gameplay.
The probability of selecting a yellow bird first and a frog second, with replacement, is \frac{3}{100}. This is calculated by multiplying the individual probabilities of drawing each card type. Specifically, the probability of a yellow bird is \frac{1}{5} and for a frog is \frac{3}{20}.
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