Probability Calculation for Last Ball Drawn
This problem involves calculating the probability of a specific outcome (the last ball drawn being black) when drawing balls without replacement.
Understanding the Drawing Process
We have a box containing:
- 40 red balls
- 60 black balls
- Total balls = 40 + 60 = 100
Balls are drawn one by one without replacement until the box is empty. We need to find the probability that the very last ball drawn is black.
Key Principle: Symmetry
In a process of drawing items randomly without replacement until none are left, every item has an equal probability of being in any specific position in the sequence of draws. This means any of the 100 balls is equally likely to be the first ball drawn, the second, ..., or the last ball drawn.
Calculating the Probability
Since each of the 100 balls has an equal chance of being the last ball drawn, we can determine the probability based on the composition of the balls:
- Total number of balls = 100
- Number of black balls = 60
The probability that the last ball drawn is black is the ratio of the number of black balls to the total number of balls.
Let B be the event that the last ball drawn is black.
$ P(B) = \frac{\text{Number of black balls}}{\text{Total number of balls}} $
$ P(B) = \frac{60}{100} $
Simplifying the fraction:
$ P(B) = \frac{6}{10} = \frac{3}{5} $
Conclusion
The probability that the last ball drawn is black is $ \frac{3}{5} $.
Important Questions from Discrete Probability
- Let $X$ be a Binomial$(n, p)$ random variable, where $n \in \{5,6\}$ and $p\in \{\frac{1}{4}, \frac{3}{4}\}$. If $X = 3$ is observed, then the maximum likelihood estimate of $(n, p)$ is
- Suppose two fair dice are thrown independently at random. Let $X$ and $Y$ be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?
Consider the problem of testing $H_0 : \theta = 1$ vs $H_1 : \theta = \frac{1}{2}$ where $\theta$ is the mean of a Poisson random variable. Let $X$ and $Y$ be a random sample from Poisson ($\theta$) distribution. Consider the following test procedure:
Reject $H_0$ if either $X = 0$ or $(X = 1 \text{ and } X + Y \leq 2)$; otherwise accept $H_0$.
Which of the following are true?
In a football league, the goals scored by home teams over 380 matches have the following frequency distribution.
Number of goals 0 1 2 3 4 5 Frequency 92 121 91 50 19 7 The average goals scored by home teams is 1.49. We want to test $H_0$: Goal distribution is Poisson. Based on observations the value of the $\chi^2$-statistic for goodness of fit is 1.27. Given $\chi^2_{0.05, 6} = 1.64, \chi^2_{0.05, 5} = 1.15, \chi^2_{0.95, 6} = 12.59$ and $\chi^2_{0.95, 5} = 11.07$, which of the following are true?
Ten balls are put in 6 slots at random. Then the expected total number of balls in the two extreme slots is
