Final answer:
To find the probability that a vowel is chosen at least times out of seven separate trials, we need to consider the different combinations of vowels that can be chosen. We divide the number of favorable outcomes by the total number of possible outcomes. After calculating the probabilities for each case, we add them together to find the overall probability.
Step-by-step explanation:
To find the probability that a vowel is chosen at least times out of seven separate trials, we need to consider the different combinations of vowels that can be chosen.
The name DAPHNE has two vowels: A and E. So, in each trial, there are two possible outcomes - either a vowel (A or E) is chosen or a consonant is chosen.
For at least four vowels, we need to consider the possibilities of choosing 4, 5, 6, or 7 vowels out of the 7 trials.
To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes.
P(Vowel at least 4 times) = P(4 vowels) + P(5 vowels) + P(6 vowels) + P(7 vowels)
P(4 vowels) = C(7,4) * (1/2)^4 * (1/2)^3
P(5 vowels) = C(7,5) * (1/2)^5 * (1/2)^2
P(6 vowels) = C(7,6) * (1/2)^6 * (1/2)^1
P(7 vowels) = C(7,7) * (1/2)^7 * (1/2)^0
After calculating the probabilities for each case, we add them together to find the overall probability.
Note: C(n,r) represents the number of combinations of choosing r items out of a set of n items, and (1/2)^k represents the probability of choosing a vowel or consonant in each trial.