A box contains 10 pink rubber bands and 20 brown rubber bands. What is the probability of picking a pink rubber band, not replacing it, and then picking another pink rubber band? P(pink and then pink)
There are a few ways to approach this problem, but one common method is to use the multiplication rule of probability. This rule states that if two events are independent, meaning that the outcome of one does not affect the outcome of the other, then the probability of both events occurring is the product of their individual probabilities.
Using this rule, we can find the probability of picking a pink rubber band on the first draw as:
P(pink on first draw) = number of pink bands / total number of bands
P(pink on first draw) = 10 / 30
P(pink on first draw) = 1/3
Since we are not replacing the first rubber band, the total number of bands for the second draw will be one less than the original total. Therefore, the probability of picking another pink rubber band on the second draw, given that the first one was pink, is:
P(pink on second draw | first was pink) = number of remaining pink bands / remaining total bands
P(pink on second draw | first was pink) = 9 / 29
To find the probability of both events occurring, we multiply these probabilities together:
P(pink and then pink) = P(pink on first draw) * P(pink on second draw | first was pink)
P(pink and then pink) = (1/3) * (9/29)
P(pink and then pink) ≈ 0.103
Therefore, the probability of picking a pink rubber band, not replacing it, and then picking another pink rubber band is approximately 0.103, or about 10.3%.
To find the probability of picking a pink rubber band, not replacing it, and then picking another pink rubber band, you need to consider the probability of the first event (picking a pink rubber band) and the probability of the second event (picking another pink rubber band).
Step 1: Calculate the probability of picking a pink rubber band on the first draw.
There are 30 rubber bands in total (10 pink + 20 brown), so the probability of picking a pink rubber band on the first draw is:
P(pink on first draw) = (number of pink rubber bands) / (total number of rubber bands)
P(pink on first draw) = 10 / 30 = 1/3
Step 2: Calculate the probability of picking another pink rubber band on the second draw without replacing the first rubber band.
After the first pink rubber band is picked, there are now 9 pink rubber bands left and 29 rubber bands in total (since one rubber band has been removed). Therefore, the probability of picking another pink rubber band is:
P(pink on second draw) = (number of remaining pink rubber bands) / (total number of remaining rubber bands)
P(pink on second draw) = 9 / 29
Step 3: Calculate the probability of both events occurring.
Since the two events are independent (the outcome of the first event does not affect the outcome of the second event), you can multiply the probabilities of the individual events to find the probability of both events occurring:
P(pink and then pink) = P(pink on first draw) * P(pink on second draw)
P(pink and then pink) = (1/3) * (9/29)
P(pink and then pink) = 9/87
Therefore, the probability of picking a pink rubber band on the first draw and then picking another pink rubber band without replacement is 9/87.