A botanist is growing 20 red roses and 40 yellow roses. The probability of selecting 10 red roses and 20 yellow roses is?
The probability of selecting 10 red roses and 20 yellow roses is 0.25 or 25%.
Well, if the botanist has a green thumb, it might increase the probability of selecting the perfect roses! But let's do some math and find out the probability.
The total number of roses is 60, and the botanist wants to select 10 red roses and 20 yellow roses.
The probability can be calculated using the formula:
(Number of ways to select 10 red roses from 20) * (Number of ways to select 20 yellow roses from 40) / (Total number of ways to select 30 roses from 60)
So, the probability is:
(20 Choose 10) * (40 Choose 20) / (60 Choose 30)
Now, if you'll excuse me, I'll need to put on my math bot hat to calculate this. Let me crunch the numbers...
*Clown Bot starts frantically typing on a calculator*
And the probability is...
[ERROR: Clown Bot's calculator overheated from too much number crunching. Please contact technical support.]
To find the probability of selecting 10 red roses and 20 yellow roses out of a total of 60 roses, we need to calculate the combination of choosing 10 red roses out of 20 and choosing 20 yellow roses out of 40.
The probability can be calculated using the formula for combinations:
P = (nCr * mCr) / NCr
where:
nCr is the combination of choosing r items from a set of n items,
mCr is the combination of choosing r items from a set of m items,
NCr is the combination of choosing r items from a set of N items,
P is the probability.
In this case, we have:
n = 20 (number of red roses)
m = 40 (number of yellow roses)
r = 10 (number of red roses chosen)
r = 20 (number of yellow roses chosen)
N = 60 (total number of roses)
Using the combination formula:
nCr = n! / (r! * (n - r)!)
mCr = m! / (r! * (m - r)!)
NCr = N! / (r! * (N - r)!)
Calculating the combinations:
nCr = 20! / (10! * (20 - 10)!)
= 20! / (10! * 10!)
= (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10!) / (10! * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
= 184,756
mCr = 40! / (20! * (40 - 20)!)
= 40! / (20! * 20!)
= (40 * 39 * 38 * 37 * 36 * 35 * 34 * 33 * 32 * 31 * 30 * 29 * 28 * 27 * 26 * 25 * 24 * 23 * 22 * 21 * 20!) / (20! * 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
= 137,846,528,820
NCr = 60! / (10! * (60 - 10)!)
= 60! / (10! * 50!)
= (60 * 59 * 58 * 57 * 56 * 55 * 54 * 53 * 52 * 51 * 50!) / (10! * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
= 3,491,432,639,508,800
Now, calculating the probability:
P = (nCr * mCr) / NCr
= (184,756 * 137,846,528,820) / 3,491,432,639,508,800
≈ 0.011024 <br>
Therefore, the probability of selecting 10 red roses and 20 yellow roses is approximately 0.011024 (or 1.1024%).
To calculate the probability of selecting a specific number of red and yellow roses, we need to know the total number of roses available. In this case, the botanist has a total of 20 red roses and 40 yellow roses, resulting in a total of 60 roses.
The probability of selecting a specific number of roses can be calculated using the concept of combinations. The formula for calculating combinations is:
C(n, r) = n! / (r!(n-r)!),
where n is the total number of objects and r is the number of objects being selected.
In this case, we want to calculate the probability of selecting 10 red roses and 20 yellow roses. So, we need to calculate the combinations for selecting 10 out of 20 red roses and 20 out of 40 yellow roses.
Step 1: Calculate the combination for selecting 10 red roses out of 20 red roses:
C(20, 10) = 20! / (10!(20-10)!) = 184,756.
Step 2: Calculate the combination for selecting 20 yellow roses out of 40 yellow roses:
C(40, 20) = 40! / (20!(40-20)!) = 137,846,528.
Step 3: Calculate the total number of possible combinations for selecting 10 red roses and 20 yellow roses:
Total Combinations = C(20, 10) * C(40, 20) = 184,756 * 137,846,528 = 25,493,889,126,208.
Step 4: Calculate the probability by dividing the favorable outcomes (the total number of combinations for selecting 10 red roses and 20 yellow roses) by the total number of possible outcomes (the total number of combinations for selecting any 30 roses out of the total 60 roses):
Probability = Total Combinations / C(60, 30).
C(60, 30) = 60! / (30!(60-30)!) = 118,264,581,564,861,424.
Probability = 25,493,889,126,208 / 118,264,581,564,861,424 ≈ 2.15 × 10^-4.
Therefore, the probability of selecting 10 red roses and 20 yellow roses from the given collection is approximately 2.15 × 10^-4.