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Let the probability of getting head for a biased coin be 1/4. It is tossed repeatedly until a head appears. Let N be the number of tosses required. If the probability that the equation 64x2 + 5Nx + 1 = 0 has no real root is p/q, where p and q are coprime, then q - p is equal to ________.
Correct answer is '27'. Can you explain this answer?
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Let the probability of getting head for a biased coin be 1/4.It is tos...
We have the quadratic equation 64x2 + 5 Nx + 1 = 0. For it to have no real roots, the discriminant (b2 - 4ac) should be less than 0. Here, a = 64, b = 5N, and c = 1.
This gives us :
Since N must be an integer (as it represents the number of tosses), the possible values of N are 1, 2, or 3.
Since N must be an integer (as it represents the number of tosses), the possible values of N are 1, 2, or 3.
The probability of getting the first head on the n- th toss (given the probability of getting a head is 1/4) is given by the geometric distribution formula, (1 - p)n-1 × p.
So, the probability for our specific values of N is:
Therefore, the total probability (p/q) is :
Therefore, q - p is equal to 27.
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Let the probability of getting head for a biased coin be 1/4.It is tos...
Finding the probability that the equation has no real roots:
Let's consider the equation 64x^2 + 5Nx + 1 = 0. For this equation to have no real roots, the discriminant (b^2 - 4ac) must be negative.
The discriminant for this equation is (5N)^2 - 4 * 64 * 1 = 25N^2 - 256.
To find the probability that the equation has no real roots, we need to find the values of N for which the discriminant is negative, and then calculate the probability of getting each of those values.
Finding the values of N for which the discriminant is negative:
For the discriminant to be negative, we have 25N^2 - 256 < />
Simplifying the inequality, we get 25N^2 < />
Dividing both sides by 25, we get N^2 < />
Taking the square root of both sides, we get N < />
N < />
Calculating the probability of getting each value of N:
Since the coin is biased and the probability of getting heads is 1/4, the probability of getting tails is 3/4.
The probability of getting N tosses with tails is (3/4)^N-1 * (1/4).
The probability of getting N tosses with heads is (3/4)^(N-1) * (1/4).
Since we are only interested in the probability of getting N tosses with tails, we can ignore the probability of getting N tosses with heads.
Therefore, the probability of getting N tosses with tails is (3/4)^N-1 * (1/4).
Calculating the probability that the equation has no real roots:
The probability that the equation has no real roots is the sum of the probabilities of getting each value of N that satisfies N < />
P(N < 16/5)="(3/4)^1" *="" (1/4)="" +="" (3/4)^2="" *="" (1/4)="" +="" (3/4)^3="" *="" (1/4)="" +="" ...="" +="" (3/4)^(16/5-1)="" *="" />
Simplifying the expression, we get:
P(N < 16/5)="(1/4)" *="" (1="" +="" 3/4="" +="" (3/4)^2="" +="" ...="" +="" />
Using the formula for the sum of a geometric series, we get:
P(N < 16/5)="(1/4)" *="" (1="" -="" (3/4)^(16/5))/(1="" -="" />
P(N < 16/5)="(1/4)" *="" (1="" -="" />
P(N < 16/5)="1" -="" />
Therefore, the probability that the equation has no real roots is 1 - (3/4)^(16/5).
Calculating q - p:
To find q - p, we subtract p from q.
q - p = 27 - 1 = 26.
Let's consider the equation 64x^2 + 5Nx + 1 = 0. For this equation to have no real roots, the discriminant (b^2 - 4ac) must be negative.
The discriminant for this equation is (5N)^2 - 4 * 64 * 1 = 25N^2 - 256.
To find the probability that the equation has no real roots, we need to find the values of N for which the discriminant is negative, and then calculate the probability of getting each of those values.
Finding the values of N for which the discriminant is negative:
For the discriminant to be negative, we have 25N^2 - 256 < />
Simplifying the inequality, we get 25N^2 < />
Dividing both sides by 25, we get N^2 < />
Taking the square root of both sides, we get N < />
N < />
Calculating the probability of getting each value of N:
Since the coin is biased and the probability of getting heads is 1/4, the probability of getting tails is 3/4.
The probability of getting N tosses with tails is (3/4)^N-1 * (1/4).
The probability of getting N tosses with heads is (3/4)^(N-1) * (1/4).
Since we are only interested in the probability of getting N tosses with tails, we can ignore the probability of getting N tosses with heads.
Therefore, the probability of getting N tosses with tails is (3/4)^N-1 * (1/4).
Calculating the probability that the equation has no real roots:
The probability that the equation has no real roots is the sum of the probabilities of getting each value of N that satisfies N < />
P(N < 16/5)="(3/4)^1" *="" (1/4)="" +="" (3/4)^2="" *="" (1/4)="" +="" (3/4)^3="" *="" (1/4)="" +="" ...="" +="" (3/4)^(16/5-1)="" *="" />
Simplifying the expression, we get:
P(N < 16/5)="(1/4)" *="" (1="" +="" 3/4="" +="" (3/4)^2="" +="" ...="" +="" />
Using the formula for the sum of a geometric series, we get:
P(N < 16/5)="(1/4)" *="" (1="" -="" (3/4)^(16/5))/(1="" -="" />
P(N < 16/5)="(1/4)" *="" (1="" -="" />
P(N < 16/5)="1" -="" />
Therefore, the probability that the equation has no real roots is 1 - (3/4)^(16/5).
Calculating q - p:
To find q - p, we subtract p from q.
q - p = 27 - 1 = 26.
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Let the probability of getting head for a biased coin be 1/4.It is tossed repeatedly until a head appears. Let Nbe the number of tosses required. If the probability that the equation 64x2 + 5Nx + 1 = 0has no real root is p/q,where pand q are coprime, then q - pis equal to ________.Correct answer is '27'. Can you explain this answer?
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