Suppose a jar contains 17 red marbles and 37 blue marbles. If you reach in the jar and pull out 2 marbles at random at the same time, find the probability that both are red.
Drawing two marbles at the same time makes a difference in the calculation of the probability. When two marbles are drawn simultaneously, the probability of each event is calculated based on the number of ways in which the event can occur divided by the total number of possible outcomes.
The total number of ways to draw 2 marbles from a jar containing 54 marbles is:
54C2 = (54 x 53) / (2 x 1) = 1431
where "C" stands for combinations, which is a way to count the number of possible combinations of objects without regard to their order.
The number of ways to draw 2 red marbles from a jar containing 17 red and 37 blue marbles is:
17C2 = (17 x 16) / (2 x 1) = 136
So, the probability of drawing 2 red marbles at the same time is:
P(both red) = 136 / 1431 = 0.095 or approximately 9.5%
Therefore, the probability of drawing 2 red marbles at the same time from the jar is 0.095 or approximately 9.5%.
The probability of drawing a red marble on the first draw is 17/(17+37) = 17/54.
After the first marble has been drawn, there will be 16 red marbles left in the jar out of a total of 53 marbles.
Therefore, the probability of drawing a second red marble is 16/53.
The probability of both events happening (drawing 2 red marbles) is equal to the product of the probabilities of each individual event:
P(both red) = P(first red) * P(second red given that the first was red)
= 17/54 * 16/53
= 0.0806 (rounded to four decimal places)
Therefore, the probability of drawing two red marbles from the jar is approximately 0.0806 or 8.06%.