MATH SOLVE

3 months ago

Q:
# A university found that 30% of its students withdraw without completing the introductory statistics course. Assume that 20 students registered for the course. Compute the probability that 2 or fewer will withdraw (to 4 decimals). Compute the probability that exactly 4 will withdraw (to 4 decimals). Compute the probability that more than 3 will withdraw (to 4 decimals). Compute the expected number of withdrawals.

Accepted Solution

A:

Step-by-step explanation:a) Compute the probability that 2 or fewer will withdrawFirst we need to determine, given 2 students from the 20. Which is the probability of those 2 to withdraw and all others to complete the course. This is given by:[tex](0.3)^2(0.7)^{18}[/tex].Then, we must multiply this quantity by [tex]{20\choose2}=\frac{20!}{18!2!}=\frac{20\times19}{2}=190,[/tex] which is the number of ways to choose 2 students from the total of 20. Therefore: the probability that exactly 2 students withdraw is [tex]190(0.3)^2(0.7)^{18}[/tex].Following an analogous process we can determine that: The probability that exactly 1 student withdraw is [tex]{20\choose1}(0.3)(0.7)^{19}=20(0.3)(0.7)^{19}.[/tex] The probability that exactly none students withdraw is [tex]{20\choose 0}(0.7)^{20}=(0.7)^{20}.[/tex]Finally, the probability that 2 or fewer students will withdraw is[tex]190(0.3)^2(0.7)^{18}+20(0.3)(0.7)^{19}+(0.7)^{20}=(0.7)^{18}(190(0.3)^2+20(0.3)(0.7)+(0.7)^2)\approx0.0355[/tex]b) Compute the probability that exactly 4 will withdraw.Following the process explained in a), the probability that 4 student withdraw is given by[tex]{20\choose4}(0.3)^4(0.7)^{16}=\frac{20\times19\times18\times17}{4\times3\times2} (0.3)^4(0.7)^{16}=4845(0.3)^4(0.7)^{16}\approx 0.1304.[/tex]c) Compute the probability that more than 3 will withdraw First we will compute the probability that exactly 3 students withdraw, which is given by[tex]{20\choose3}(0.3)^3(0.7)^{17}=\frac{20\times19\times18}{3\times2} (0.3)^3(0.7)^{17}=1140(0.3)^3(0.7)^{17}\approx 0.0716.[/tex]Then, using a) we have that the probability that 3 or fewer students withdraw is 0.0355+0.0716=0.1071. Therefore the probability that more than 3 will withdraw is 1-0.1071=0.8929d) Compute the expected number of withdrawals.As stated in the problem, 30% of the students withdraw, then, the expected number of withdrawals is the 30% of 20 which is 6.