According to genetic theory, every plant of a particular species has a 25% chance of being red-flowering, independently of all other plants. Among 10 plants of this species, what is the chance that fewer than 4 are red-flowering?
0.7758
binomial distribution:
p=0.25
n=10
X<=3
To find the chance that fewer than 4 plants are red-flowering, we need to consider each possible outcome.
Let's calculate the probability for each case:
Case 0: 0 red-flowering plants.
The probability of a plant being red-flowering is 25%, so the probability of a plant not being red-flowering is 75% (1 - 0.25).
Using the binomial probability formula, the probability of all 10 plants not being red-flowering is (0.75)^10 = 0.0563.
Case 1: 1 red-flowering plant.
We have 10 possible positions for the red-flowering plant, so we multiply the probability of having one plant red-flowering by 10.
The probability of having 1 red-flowering plant is (0.25)^1 * (0.75)^9, and the probability of occurring in any of the 10 positions is 10 * (0.25)^1 * (0.75)^9 = 0.2503.
Case 2: 2 red-flowering plants.
Similarly, the probability of having 2 red-flowering plants is (0.25)^2 * (0.75)^8, and since there are different combinations for which plants can be red-flowering, we use the binomial coefficient.
The binomial coefficient for choosing 2 plants out of 10 is calculated as (10 choose 2) = 10! / (2! * (10-2)!) = 45.
So the probability of having 2 red-flowering plants is 45 * (0.25)^2 * (0.75)^8 = 0.2637.
Case 3: 3 red-flowering plants.
The probability of having 3 red-flowering plants is (0.25)^3 * (0.75)^7.
Using the binomial coefficient again, (10 choose 3) = 10! / (3! * (10-3)!) = 120.
So the probability of having 3 red-flowering plants is 120 * (0.25)^3 * (0.75)^7 = 0.3955.
Now, we sum up the probabilities of all these cases to get the final probability:
Probability of fewer than 4 red-flowering plants = Probability of Case 0 + Probability of Case 1 + Probability of Case 2 + Probability of Case 3
= 0.0563 + 0.2503 + 0.2637 + 0.3955
= 0.9658
Therefore, the chance that fewer than 4 plants are red-flowering among the 10 plants of this species is approximately 0.9658, or 96.58%.