A container contains two red, three yellow, and four green cards. Two cards are selected at random, one after the other and without replacement, from the container. Calculate the probability that one card is red and one card is yellow.
To calculate the probability that one card is red and one card is yellow, we need to determine the total number of possible outcomes and the number of favorable outcomes.
First, let's calculate the total number of possible outcomes:
There are 9 cards in total (2 red + 3 yellow + 4 green).
For the first card, we have 9 options. After selecting one card, there are 8 cards left for the second selection.
Therefore, the total number of possible outcomes is 9 * 8 = 72.
Next, let's determine the number of favorable outcomes:
To have one card red and one card yellow, we can choose a red card first and then a yellow card or vice versa.
For the first selection, we have 2 red cards to choose from. After selecting one red card, there are 3 yellow cards left for the second selection. So we have 2 * 3 = 6 favorable outcomes.
Alternatively, we could first select a yellow card then a red card. Again, we have 3 options for the first selection. After selecting one yellow card, there are 2 red cards remaining. So there are 3 * 2 = 6 favorable outcomes.
Adding these two possibilities together, we have a total of 6 + 6 = 12 favorable outcomes.
Finally, let's calculate the probability:
The probability is given by the number of favorable outcomes divided by the total number of possible outcomes.
Therefore, the probability that one card is red and one card is yellow is 12/72, which simplifies to 1/6.
So the probability is 1/6 or 0.1667 (rounded to four decimal places).
To calculate the probability, we need to find the number of favorable outcomes (i.e., selecting one red card and one yellow card) and the total number of possible outcomes.
First, let's find the total number of possible outcomes:
There are 9 cards in total, and we are selecting 2 cards without replacement. Therefore, the total number of possible outcomes is the number of ways to choose 2 cards from 9, which can be calculated using combinations.
The number of combinations of selecting 2 cards from 9 is given by:
C(9, 2) = 9! / (2! * (9 - 2)!) = 9! / (2! * 7!) = (9 * 8) / (2 * 1) = 36.
Now, let's find the number of favorable outcomes:
We need to choose one red card and one yellow card.
The number of ways to choose 1 red card from 2 is given by:
C(2, 1) = 2! / (1! * (2 - 1)!) = 2.
Similarly, the number of ways to choose 1 yellow card from 3 is given by:
C(3, 1) = 3! / (1! * (3 - 1)!) = 3.
Therefore, the number of favorable outcomes is 2 * 3 = 6.
Finally, the probability of selecting one red card and one yellow card is:
Probability = Favorable outcomes / Total outcomes = 6 / 36 = 1 / 6.
So, the probability that one card is red and one card is yellow is 1/6 or approximately 0.1667.
To calculate the probability that one card is red and one card is yellow, we need to determine the total number of possible outcomes and the number of favorable outcomes.
There are nine cards in total (2 red + 3 yellow + 4 green). The first card can be any of the nine cards. After removing one card, there are eight cards remaining, so the second card can be chosen from the remaining eight cards.
Now we need to determine the number of favorable outcomes. Since we want one red card and one yellow card, there are two possible cases:
1. Selecting a red card first and then a yellow card.
2. Selecting a yellow card first and then a red card.
Let's calculate each case separately:
Case 1:
The probability of selecting a red card first is 2/9 (since there are 2 red cards out of 9 total cards).
After removing one red card, there are 8 cards remaining, out of which 3 are yellow. So the probability of selecting a yellow card after selecting a red card is 3/8.
Therefore, the probability of getting a red card first and then a yellow card is (2/9) * (3/8) = 6/72.
Case 2:
The probability of selecting a yellow card first is 3/9 (since there are 3 yellow cards out of 9 total cards).
After removing one yellow card, there are 8 cards remaining, out of which 2 are red. So the probability of selecting a red card after selecting a yellow card is 2/8.
Therefore, the probability of getting a yellow card first and then a red card is (3/9) * (2/8) = 6/72.
Since both cases are independent events, we can add their probabilities together:
(6/72) + (6/72) = 12/72 = 1/6.
Hence, the probability that one card is red and one card is yellow is 1/6.