Sally has a jar of 10 marbles, 3 red, 4 blue, and 3 green.
If Sally reaches into her jar and selects two marbles at random, what is the probability that the first marble is red and the second is also red? What is the probability that the first is red and the second is blue?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
Assuming there is no replacement:
P(red,red) = 3/10 * 2/9 = ?
P(red,blue) = 3/10 * 4/9 = ?
To find the probability, we need to determine the total number of possible outcomes and the number of favorable outcomes for each scenario.
Let's start with the probability of selecting two red marbles consecutively.
Step 1: Finding the total number of possible outcomes
Sally is selecting two marbles from a jar that contains a total of 10 marbles. So, there are 10 choices for the first marble and 9 choices for the second one. However, since order matters (we're looking for consecutive red marbles), we also need to consider the number of ways the two red marbles can be arranged. Therefore, the total number of possible outcomes is 10 * 9 = 90.
Step 2: Finding the number of favorable outcomes
Since there are 3 red marbles in the jar, Sally has 3 choices for the first marble and 2 choices for the second one (as the number of red marbles decreases by 1). Therefore, the number of favorable outcomes is 3 * 2 = 6.
Step 3: Calculating the probability
The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability of selecting two red marbles consecutively = 6 / 90 = 1 / 15 ≈ 0.0667 or 6.67%
Now, let's move on to calculating the probability of selecting a red marble first and a blue marble second.
Step 1: Finding the total number of possible outcomes
The total number of possible outcomes remains the same at 90 since we have not replaced the marbles after the first selection.
Step 2: Finding the number of favorable outcomes
Sally has 3 choices for the first red marble and 4 choices for the second blue marble (as there are 4 blue marbles in the jar). Therefore, the number of favorable outcomes is 3 * 4 = 12.
Step 3: Calculating the probability
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability of selecting a red marble first and a blue marble second = 12 / 90 = 2 / 15 ≈ 0.1333 or 13.33%
Therefore, the probability that the first marble is red and the second is also red is approximately 6.67%, and the probability that the first marble is red and the second is blue is approximately 13.33%.