Answer:
(a)
(i) The probability of getting at least one 3 when 9 fair dice are thrown is 0.8062.
(ii) The value of <em>n</em> is 12.
(b) The probability that Ronnie wins the game is 0.3572.
Step-by-step explanation:
(a)
(i)
The probability of getting a 3 on a single die roll is, 
.
It is provided that <em>n</em> = 9 fair dice are thrown together.
The outcomes of each die is independent of the others.
The random variable <em>X</em> can be defined as the number of die with outcome as 3.
The random variable <em>X</em> follows a Binomial distribution with parameters <em>n</em> = 9 and .
Compute the probability of getting at least one 3 as follows:
![=1-[{9\choose 0}(\frac{1}{6})^{0}(1-\frac{1}{6})^{9-0}]\\\\=1-(\frac{5}{6})^{9}\\\\=1-0.19381\\\\=0.80619\\\\\approx 0.8062](https://tex.z-dn.net/?f=%3D1-%5B%7B9%5Cchoose%200%7D%28%5Cfrac%7B1%7D%7B6%7D%29%5E%7B0%7D%281-%5Cfrac%7B1%7D%7B6%7D%29%5E%7B9-0%7D%5D%5C%5C%5C%5C%3D1-%28%5Cfrac%7B5%7D%7B6%7D%29%5E%7B9%7D%5C%5C%5C%5C%3D1-0.19381%5C%5C%5C%5C%3D0.80619%5C%5C%5C%5C%5Capprox%200.8062)
Thus, the probability of getting at least one 3 when 9 fair dice are thrown is 0.8062.
(ii)
It is provided that:
P (X ≥ 1) > 0.90
Compute the value of <em>n</em> as follows:
Thus, the value of <em>n</em> is 12.
(b)
It is provided that the bag contains 5 green balls and 3 yellow balls.
Ronnie and Julie play a game in which they take turns to draw a ball from the bag at random without replacement.
The winner of the game is the first person to draw a yellow ball.
Also provided that Julie draws the first ball.
P (Ronnie Wins) = P (The 1st yellow ball is selected at an even draw)
= P (The 1st yellow ball is drawn at 2nd, 4th and 6th draw)
= P (1st yellow ball is drawn at 2nd)
+ P (1st yellow ball is drawn at 4th)
+ P (1st yellow ball is drawn at 6th)
![=[\frac{5}{8}\times \frac{3}{7}]+[\frac{5}{8}\times \frac{3}{7}\times \frac{4}{6}\times \frac{2}{5}]+[\frac{5}{8}\times \frac{3}{7}\times \frac{4}{6}\times \frac{2}{5}\times \frac{1}{4}\times 1]\\\\=0.2679+0.0714+0.0179\\\\=0.3572](https://tex.z-dn.net/?f=%3D%5B%5Cfrac%7B5%7D%7B8%7D%5Ctimes%20%5Cfrac%7B3%7D%7B7%7D%5D%2B%5B%5Cfrac%7B5%7D%7B8%7D%5Ctimes%20%5Cfrac%7B3%7D%7B7%7D%5Ctimes%20%5Cfrac%7B4%7D%7B6%7D%5Ctimes%20%5Cfrac%7B2%7D%7B5%7D%5D%2B%5B%5Cfrac%7B5%7D%7B8%7D%5Ctimes%20%5Cfrac%7B3%7D%7B7%7D%5Ctimes%20%5Cfrac%7B4%7D%7B6%7D%5Ctimes%20%5Cfrac%7B2%7D%7B5%7D%5Ctimes%20%5Cfrac%7B1%7D%7B4%7D%5Ctimes%201%5D%5C%5C%5C%5C%3D0.2679%2B0.0714%2B0.0179%5C%5C%5C%5C%3D0.3572)
Thus, the probability that Ronnie wins the game is 0.3572.
