Let $\Omega_1$, $\Omega_2$, and $\Omega_3$ be three urns and each urn contains 5 numbered balls respectively 1, 2, 3, 4, and 5. One ball is drawn at random from each urn, all the balls from each urn having the same probability of being drawn.
- Evaluate the probability that the number 5 does not come out in the draw.
- Evaluate the probability that the three numbers drawn are less than or equal to 3.
- Evaluate the probability that the largest number in the draw is a 4.
My answers are :
- $P(1) = (\frac{4}{5})^3$
- $P(2) = (\frac{3}{5})^3$
- $P(3) = (\frac{1}{5}) \cdot (\frac{4}{5})^2 = 0.128$
are my answers right ?
Best Answer
Your first two answers are correct, but the third is not.
It looks like what you have calculated is: $$P(\text{first ball is 4 & second ball is $\leq4$ & third ball is $\leq 4$}).$$ Certainly these outcomes will result in the largest draw being $4$, but there are other outcomes you have missed. For instance, it might be the second or third ball that is $4$. However, we can't just add those probabilities on to what you calculated, because then we are overcounting - the outcome $(4,4,4)$ gets counted in all three cases, for instance.
It starts to get complicated keeping track of all the ways the largest draw can be $4$ without overcounting. The easier way is to think about writing this event in terms of other events whose probabilities are easier to calculate. In particular, we can write $$\{\text{largest draw is $4$}\} = \{\text{all draws are $\leq 4$}\} \setminus \{\text{all draws are $\leq 3$}\}.$$ This is because the largest draw is $4$ if and only if all the draws are $\leq 4$ and not all the draws are $\leq 3$. Then since $\{\text{all draws are $\leq 3$}\} \subseteq \{\text{all draws are $\leq 4$}\}$, we have: $$P(\text{largest ball is $4$}) = P(\text{all balls are $\leq 4$}) - P(\text{all balls are $\leq 3$}).$$ Finally, you can calculate $$P(\text{all balls are $\leq k$}) = P(\text{first ball is $\leq k$})\cdot P(\text{second ball is $\leq k$}) \cdot P(\text{third ball is $\leq k$})$$ for $k=3,4$.