from a batch of 10 eggs 3 are rotten.2 eggs are selected in succession.Find the probability that(a)both are rotten (b)only 1 is rottewn and at least 1 is fresh.
b) Probability(1rotten, 1fresh)=3/10*7/9
Notice how the probabality after drawing one rotten, the fresh egg probabality increases slightly.
To calculate the probability of selecting eggs from a batch, we can use the concept of conditional probability. Let's break down the problem step by step:
a) Probability of both eggs being rotten:
In this case, we have 3 rotten eggs out of a total of 10. Since we are selecting 2 eggs in succession without replacement, the sample space decreases after each selection.
The probability of selecting the first rotten egg is 3/10, as there are 3 rotten eggs out of the total 10.
After removing one rotten egg, there are 2 rotten eggs remaining out of 9 eggs.
The probability of selecting the second rotten egg, given that the first egg was rotten, is 2/9.
To find the probability of both eggs being rotten, we multiply the individual probabilities:
Probability(both are rotten) = (3/10) * (2/9) = 6/90 = 1/15
b) Probability of 1 rotten and at least 1 fresh egg:
In this case, we need to consider two scenarios:
(i) The first egg is rotten and the second egg is fresh.
(ii) The first egg is fresh and the second egg is rotten.
For scenario (i), the probability of the first egg being rotten is still 3/10. After removing one rotten egg, we have 7 fresh eggs remaining out of a total of 9.
The probability of selecting a fresh egg as the second one, given that the first egg was rotten, is 7/9.
So, for scenario (i), the probability is: (3/10) * (7/9).
For scenario (ii), the probability of the first egg being fresh is 7/10 (since 7 eggs are fresh out of a total of 10). After removing one fresh egg, we have 2 rotten eggs remaining out of 9.
The probability of selecting a rotten egg as the second one, given that the first egg was fresh, is 2/9.
So, for scenario (ii), the probability is: (7/10) * (2/9).
To find the probability of only 1 rotten and at least 1 fresh egg, we add the probabilities of both scenarios:
Probability(1 rotten, at least 1 fresh) = (3/10) * (7/9) + (7/10) * (2/9)
Simplifying the expression gives us the final probability.