Final answer:
a. x has a geometric distribution. b. The probability that it will take exactly two tosses for Sophie to catch a ball is 0.09. c. The probability that more than three tosses will be required is 0.729. d. The expected time until Sophie catches the ball is 10 tosses and the standard deviation is 3.
Step-by-step explanation:
a. The random variable x represents the number of tosses required until Sophie catches a ball. Since each toss is an independent trial with a fixed probability of success (0.1), x follows a geometric distribution.
b. To find the probability that it will take exactly two tosses for Sophie to catch a ball, we can use the formula for the geometric distribution: P(x = k) = (1-p)^(k-1) * p, where p is the probability of success. Substituting the given values, we have P(x = 2) = (0.9)^(2-1) * 0.1 = 0.09.
c. The probability that more than three tosses will be required is equal to 1 minus the sum of the probabilities of catching the ball on the first three tosses. P(x > 3) = 1 - P(x = 1) - P(x = 2) - P(x = 3) = 1 - 0.1 - 0.09 - 0.081 = 0.729.
d. The expected time and standard deviation until Sophie catches the ball can be calculated using the formulas for the geometric distribution. The expected time (mean) is given by E(x) = 1/p, where p is the probability of success. In this case, E(x) = 1/0.1 = 10 tosses. The standard deviation is given by SD(x) = sqrt((1-p)/p^2), which in this case is SD(x) = sqrt((0.9)/0.1^2) = 3.}