Suppose a girl throws a die. If she gets 1 or 2 , she tosses a coin three times and notes the number of tails. If she gets 3 , 4 , 5 , or 6 , she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3 , 4 , 5 , or 6 with the die ?

Let E 1 be the event that the girl gets 1 or 2 . Thus, P E 1 = 2 6 = 1 3 Let E 2 be the event that the girl gets 3 , 4 , 5 , or 6 . Then P E 2 = 4 6 = 2 3 Let A be the event that she obtained exactly one tail. If she tossed a coin 3 times and exactly 1 tail showed up, then the total number of favorable outcomes = { ( THH ) , ( HTH ) , ( HHT ) } = 3 Thus, P A ∣ E 1 = 3 8 If she tossed the coin only once and exactly 1 tail showed up, the total number of favorable outcomes = T = 1 . Thus, P A ∣ E 2 = 1 2 The probability that she threw 3 , 4 , 5 or 6 with the die, given that she got exactly one tail can be found as follows: Or in other words, we need to find P E 2 | A We can use the Baye's theorem, according to which P E 2 | A = P E 2 P A ∣ E 2 P E 1 P A ∣ E 1 + P E 2 P A ∣ E 2 = 1 2 · 2 3 1 2 · 2 3 + 3 8 · 1 3 = 1 3 1 3 + 1 8 = 8 11 Hence, the required probability is 8 11 .

E M B I B E

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 13 - Probability>Exercise>Q 51

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IMPORTANT

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A die is thrown twice and the sum of the numbers appearing is observed to be $6$ . What is the conditional probability that the number $4$ has appeared at least once ?

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Important Questions on Probability

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Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 13 - Probability>Exercise>Q 52

A card from a pack of

52

cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

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Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 13 - Probability>Exercise>Q 53

Bag I contains

3

red and

4

black balls while Bag II contains

4

red and

5

black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn form Bag II and it was found to be red. Find the probability that the transferred ball is black.

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Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 13 - Probability>Exercise>Q 54

If $X$ is a random variable such that $σ (X) = 1.5$ , then, $σ (- 3 X + 1) =$