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A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q. If p: q = m : n, where m and n are coprime, then m + n is equal to:
Correct answer is '14'. Can you explain this answer?
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A bag contains six balls of different colours. Two balls are drawn in ...
Probability of drawing two balls of the same color:
Let's consider the possible outcomes for the first draw. Since there are six balls of different colors, the probability of drawing any particular color is 1/6. Therefore, the probability of drawing two balls of the same color on the first draw is:
P(same color on first draw) = (1/6) * (1/6) = 1/36
Since the balls are replaced after each draw, the probability of drawing the same color on the second draw is also 1/36. Therefore, the probability of drawing two balls of the same color in succession is:
P(both balls of the same color) = P(same color on first draw) * P(same color on second draw) = (1/36) * (1/36) = 1/1296
Probability of drawing exactly three balls of the same color:
Now let's consider the possible outcomes for the first four draws. Since there are six balls of different colors, the probability of drawing any particular color is 1/6. Therefore, the probability of drawing three balls of the same color and one ball of a different color on the first four draws is:
P(three balls of the same color) = (1/6)^3 * (5/6) = 5/1296
Since the balls are replaced after each draw, the probability of drawing a different color on the fifth draw is 1 - (1/6) = 5/6. Therefore, the probability of drawing exactly three balls of the same color in succession is:
P(exactly three balls of the same color) = P(three balls of the same color) * P(different color on fifth draw) = (5/1296) * (5/6) = 25/7776
Calculating the ratio:
Now we can calculate the ratio of p to q:
p: q = P(both balls of the same color) : P(exactly three balls of the same color)
= 1/1296 : 25/7776
= 6/7776 : 25/7776
= 6 : 25
Since m and n are coprime (meaning they have no common factors other than 1), the numerator and denominator of the ratio are already in their simplest form. Therefore, m = 6 and n = 25.
To find the product m * n, we simply multiply the values:
m * n = 6 * 25 = 150
Therefore, m * n = 150, not 14 as stated in the question.
Let's consider the possible outcomes for the first draw. Since there are six balls of different colors, the probability of drawing any particular color is 1/6. Therefore, the probability of drawing two balls of the same color on the first draw is:
P(same color on first draw) = (1/6) * (1/6) = 1/36
Since the balls are replaced after each draw, the probability of drawing the same color on the second draw is also 1/36. Therefore, the probability of drawing two balls of the same color in succession is:
P(both balls of the same color) = P(same color on first draw) * P(same color on second draw) = (1/36) * (1/36) = 1/1296
Probability of drawing exactly three balls of the same color:
Now let's consider the possible outcomes for the first four draws. Since there are six balls of different colors, the probability of drawing any particular color is 1/6. Therefore, the probability of drawing three balls of the same color and one ball of a different color on the first four draws is:
P(three balls of the same color) = (1/6)^3 * (5/6) = 5/1296
Since the balls are replaced after each draw, the probability of drawing a different color on the fifth draw is 1 - (1/6) = 5/6. Therefore, the probability of drawing exactly three balls of the same color in succession is:
P(exactly three balls of the same color) = P(three balls of the same color) * P(different color on fifth draw) = (5/1296) * (5/6) = 25/7776
Calculating the ratio:
Now we can calculate the ratio of p to q:
p: q = P(both balls of the same color) : P(exactly three balls of the same color)
= 1/1296 : 25/7776
= 6/7776 : 25/7776
= 6 : 25
Since m and n are coprime (meaning they have no common factors other than 1), the numerator and denominator of the ratio are already in their simplest form. Therefore, m = 6 and n = 25.
To find the product m * n, we simply multiply the values:
m * n = 6 * 25 = 150
Therefore, m * n = 150, not 14 as stated in the question.
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A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q.If p: q = m : n,where m and n arecoprime, then m + n is equal to:Correct answer is '14'. Can you explain this answer?
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A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q.If p: q = m : n,where m and n arecoprime, then m + n is equal to:Correct answer is '14'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q.If p: q = m : n,where m and n arecoprime, then m + n is equal to:Correct answer is '14'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q.If p: q = m : n,where m and n arecoprime, then m + n is equal to:Correct answer is '14'. Can you explain this answer?.
A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q.If p: q = m : n,where m and n arecoprime, then m + n is equal to:Correct answer is '14'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q.If p: q = m : n,where m and n arecoprime, then m + n is equal to:Correct answer is '14'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is p. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is q.If p: q = m : n,where m and n arecoprime, then m + n is equal to:Correct answer is '14'. Can you explain this answer?.
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