Twenty marbles are in a bag. There are 5 green marbles, 9 blue marbles, and the rest are red. Two marbles are drawn at random without placement. Find the probability that neither is blue
To find the probability that neither marble is blue, we need to determine the number of favorable outcomes (marbles that are not blue) and the total number of possible outcomes.
There are a total of 20 marbles in the bag, with 5 green, 9 blue, and the rest red. Since the number of red marbles is not specified, we need to calculate it.
The number of red marbles = Total marbles − (Number of green marbles + Number of blue marbles)
The number of red marbles = 20 − (5 + 9)
The number of red marbles = 20 − 14
The number of red marbles = 6
Now, let's calculate the number of favorable outcomes.
To have neither marble blue, both marbles drawn must be green or red.
Number of favorable outcomes = Number of green marbles + Number of red marbles
Number of favorable outcomes = 5 + 6
Number of favorable outcomes = 11
Finally, let's calculate the total number of possible outcomes.
Since two marbles are drawn without replacement, the total number of possible outcomes is the number of ways to choose 2 marbles out of 20.
Total number of possible outcomes = Combination(20, 2)
Total number of possible outcomes= 20! / (2!(20 - 2)!)
Total number of possible outcomes = (20 * 19) / (2 * 1)
Total number of possible outcomes = 190
Therefore, the probability that neither marble is blue is:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 11 / 190
Probability ≈ 0.058 (rounded to three decimal places)
So, the probability that neither marble is blue is approximately 0.058.
To find the probability that neither of the two marbles drawn is blue, we first need to determine the number of red marbles in the bag.
Given that there are 5 green marbles and 9 blue marbles, we can subtract the sum of these two from the total number of marbles in the bag to find the number of red marbles:
Total marbles in the bag = green marbles + blue marbles + red marbles
Total marbles in the bag = 5 + 9 + red marbles
Since the total number of marbles in the bag is given as 20, we can set up the equation:
20 = 5 + 9 + red marbles
To solve for red marbles, we subtract 5 and 9 from both sides of the equation:
20 - 5 - 9 = red marbles
6 = red marbles
Therefore, there are 6 red marbles in the bag.
Now, let's find the probability of not selecting a blue marble for the first draw. The probability of picking a red marble on the first draw is:
Probability of red on first draw = (number of red marbles) / (total number of marbles)
Probability of red on first draw = (6) / (20)
Since we do not replace the marble after it is drawn, there is now one fewer marble in the bag for the second draw. Consequently, the probability of not selecting a blue marble on the second draw is:
Probability of red on second draw = (number of red marbles) / (total number of remaining marbles)
Probability of red on second draw = (6) / (19)
To find the probability of neither draw being blue, we multiply the probabilities of these two events together:
Probability of neither being blue = Probability of red on first draw * Probability of red on second draw
Probability of neither being blue = (6 / 20) * (6 / 19)
Simplifying this expression gives us the final probability:
Probability of neither being blue ≈ 0.0947 or 9.47% (rounded to the nearest hundredth)