Question
There are three boxes. First box has 2 white, 3 black and 4 red balls. Second box has 3 white, 2 black and 2 red balls. Third box has 4 white, 1 black and 3 red balls. A box is chosen at random and 2 balls are drawn out of which 1 is white, and 1 is red. What is the probability that the balls came from first box?
The correct answer is
0.237
Let E mean drawing one white and one red ball without replacement. For the three boxes, \(P(E\mid B_1)=\frac{2\cdot4}{\binom92}=\frac29\), \(P(E\mid B_2)=\frac{3\cdot2}{\binom72}=\frac27\), and \(P(E\mid B_3)=\frac{4\cdot3}{\binom82}=\frac37\). The equally likely box choices cancel in Bayes’ formula: \(P(B_1\mid E)=\frac{2/9}{2/9+2/7+3/7}=\frac{14}{59}\approx0.237\).
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Important Questions from Probability
- If one event is unaffected by the outcome of another event, the two events are said to be:
1. Dependent
2. Independent
3. Mutually exclusive
4. Mutually inclusive - Let $M$ be a randomly chosen non-empty subset of $S = \{1, 2, 3, \dots, 2026\}$.
Which of the following is the probability that the product of all the elements of $M$ is even? - A fair coin is tossed 6 times. The probability of getting at least 5 heads is _________________. (rounded off to two decimal places)
- Consider a random experiment of rolling a fair die. The probability of getting an odd number or a number less than 4 is ______ (Rounded off to two decimal places).
- In a null hypothesis significance test, the significance level $\alpha$ was set to 0.01. For which one or more of the following $p$ values would the null hypothesis be rejected?
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