There are 7 rap songs, 7 rock songs and 5 classical songs to select from. Three songs are to be chosento play at random
-what is the probability that only classical songs are selected?
-what is the expected number of classical songs to be selected?
number of all possible triples selected = C(19,3) = 969
number with exactly one classical, 2 from the others
= C(5,1)xC(14,2) = 5(91) = 455
prob of your event = 455/969
don't understand your answer Reiny
To find the probability of only classical songs being selected, we need to calculate the number of favorable outcomes (only classical songs) and the total number of possible outcomes.
The total number of possible outcomes is the combination of selecting 3 songs out of the total number of songs available, which is (7+7+5) = 19. We can calculate this using the formula for combinations:
Total possible outcomes = C(19, 3) = 19! / (3! * (19-3)!) = 19! / (3! * 16!) = (19 * 18 * 17) / (3 * 2 * 1) = 969
Now, let's calculate the number of favorable outcomes (only classical songs). Since there are 5 classical songs in total, we can select all 3 songs from the classical category.
Number of favorable outcomes = C(5, 3) = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = (5 * 4 * 3) / (3 * 2 * 1) = 10
So, the probability of only classical songs being selected is:
Probability = Number of favorable outcomes / Total possible outcomes = 10 / 969 ≈ 0.0103
Now let's calculate the expected number of classical songs to be selected.
The expected value is calculated by multiplying each possible outcome by its probability and then summing them up. Here, we have two possible outcomes: either all three songs are classical (0 classical songs selected = 0) or some classical songs are selected (1, 2, or 3 classical songs are selected).
The probability of selecting 0 classical songs is the complement of the probability of only selecting classical songs:
P(0 classical songs) = 1 - P(only classical songs) = 1 - 0.0103 = 0.9897
The expected number of classical songs to be selected is:
Expected value = 0 * P(0 classical songs) + 1 * P(1 classical song) + 2 * P(2 classical songs) + 3 * P(3 classical songs)
Here, P(1 classical song), P(2 classical songs), and P(3 classical songs) can be calculated using similar combinations as above, considering the different possibilities of selecting songs from each category.
After calculating these probabilities, you can multiply the respective probabilities with the number of classical songs to get:
Expected value = 0 * 0.9897 + 1 * P(1 classical song) + 2 * P(2 classical songs) + 3 * P(3 classical songs)
I'll leave the calculations for P(1 classical song), P(2 classical songs), and P(3 classical songs) up to you, as they will require similar calculations using combinations as shown above.