(1) A bag contains 3 red marbles and 5 blue marbles. Jake will choose 2 marbles without looking. What is the probability that he will choose a blue marble followed by a red marble?
(2) Suppose Jake wants to pick a third marble from the set in question 1. He has already picked a blue marble and then a red one . What is the probability that the third marble will be red?
1. Blue = 5/(3+5) = ?
Since blue marble is already gone, red = 3/(3+4) = ?
Probability of all events occurring is found by multiplying the probabilities of the individual events.
2. 2/(2+4) = ?
To solve both of these probability questions, we can use the concept of conditional probability. Conditional probability involves finding the probability of an event happening given that another event has already occurred.
Let's answer each question step by step:
(1) To find the probability that Jake will choose a blue marble followed by a red marble, we need to first find the probability of picking a blue marble and multiply it by the probability of picking a red marble, given that a blue marble has already been picked.
Step 1: Calculate the probability of picking a blue marble.
The bag contains a total of 8 marbles (3 red + 5 blue). To find the probability of picking a blue marble, we divide the number of blue marbles by the total number of marbles:
P(blue) = 5/8
Step 2: Calculate the probability of picking a red marble, given that a blue marble has already been picked.
After Jake picks a blue marble, there are now 7 marbles left in the bag (3 red + 4 blue). To find the probability of picking a red marble after a blue one has been picked, we divide the number of red marbles by the total number of remaining marbles:
P(red|blue) = 3/7
Step 3: Multiply the probabilities calculated in steps 1 and 2 to get the final probability:
P(blue followed by red) = P(blue) * P(red|blue)
P(blue followed by red) = (5/8) * (3/7)
P(blue followed by red) = 15/56
Therefore, the probability that Jake will choose a blue marble followed by a red marble is 15/56.
(2) To find the probability that the third marble will be red, given that Jake has already picked a blue marble followed by a red marble, we need to calculate the probability of picking a red marble from the remaining marbles.
Step 1: Calculate the number of remaining marbles after Jake's previous picks.
After Jake picks a blue marble followed by a red marble, there is now one less blue marble and one less red marble in the bag. Therefore, there are 2 red marbles and 4 blue marbles left.
Step 2: Calculate the probability of picking a red marble from the remaining marbles.
To find the probability that the third marble will be red, we divide the number of red marbles (2) by the total number of remaining marbles (2 red + 4 blue):
P(red) = 2/6
Therefore, the probability that the third marble will be red, given that Jake has already picked a blue marble followed by a red marble, is 2/6, which simplifies to 1/3.