a box contains 25 computer diskettes, 5 of which are double density and the remiander are high density. A second box contains 15 diskettes, 10 double density and 5 high density. A box is selected at random and then two diskettes are chosen randomly from the selected box. What is the probability that both diskettes are high density?
The probabality of selecting a HD is
(20+5)/(25+15) * (24)/(39)
There are longer and more complicated of working this out.
can you explain?
To find the probability that both diskettes are high density, we need to calculate the probability of two events happening: (1) selecting a box that contains high-density diskettes, and (2) selecting two high-density diskettes from that box.
Let's start by determining the probability of selecting a box that contains high-density diskettes.
There are two boxes to choose from: Box 1 and Box 2. Box 1 contains 25 diskettes with 5 double density and the remainder high density. Box 2 contains 15 diskettes with 10 double density and 5 high density.
Since we are choosing a box at random, the probability of selecting Box 1 is 1/2, and the probability of selecting Box 2 is also 1/2.
Now, let's move on to the probability of selecting two high-density diskettes from the chosen box.
If we select Box 1, which contains 25 diskettes, the probability of selecting a high-density diskette on the first draw is 20/25, or 4/5. After the first diskette is chosen, there are still 19 high-density diskettes remaining out of the remaining 24 total diskettes. So, the probability of selecting another high-density diskette on the second draw is 19/24.
If we select Box 2, which contains 15 diskettes, the probability of selecting a high-density diskette on the first draw is 5/15, or 1/3. After the first diskette is chosen, there are still 4 high-density diskettes remaining out of the remaining 14 total diskettes. So, the probability of selecting another high-density diskette on the second draw is 4/14, which simplifies to 2/7.
To calculate the overall probability, we need to consider the cases when we choose either Box 1 or Box 2:
Probability of selecting Box 1 and both high-density diskettes:
(1/2) * (4/5) * (19/24) = 38/80
Probability of selecting Box 2 and both high-density diskettes:
(1/2) * (1/3) * (2/7) = 2/42
Now, let's sum up the probabilities from both cases:
(38/80) + (2/42) = 38/80 + 4/84 = 42/84
Simplifying the fraction, we get:
42/84 = 1/2
Therefore, the probability that both diskettes chosen are high density is 1/2.