In a random experiment, two dice are thrown and the sum of the numbers appeared on them is recorded. This experiment is repeated 9 times. If the probability that a sum of 6 appears at least once is \( P_1 \) and a sum of 8 appears at least once is \( P_2 \), then \( P_1 : P_2 = \):
Show Hint
- \( 4:3 \)
- \( 3:1 \)
- \( 1:2 \)
- \( 1:1 \)
The Correct Option is D
Approach Solution - 1
Approach Solution -2
Given that two dice are thrown, the sum of numbers appearing can be any integer from 2 to 12. Let's determine the probability for specific sums.
The pairs that give a sum of 6 are: (1,5), (2,4), (3,3), (4,2), (5,1). Thus, there are 5 successful outcomes. The sample space when throwing two dice is 6×6=36. Therefore, the probability \( P(S=6) \) is \( \frac{5}{36} \).
The pairs that give a sum of 8 are: (2,6), (3,5), (4,4), (5,3), (6,2). Thus, there are also 5 successful outcomes. Therefore, the probability \( P(S=8) \) is also \( \frac{5}{36} \).
The complement probability is used to calculate the probability of the event occurring at least once: If \( A \) is the event that a sum appears at least once in 9 trials, then the complement event \( A' \) is that the sum does not appear in all 9 trials.
Let \( \bar{P_1} \) be the probability that a sum of 6 does not appear in a single trial, so \( \bar{P_1} = 1 - \frac{5}{36} = \frac{31}{36} \). Thus, the probability that 6 does not appear in all 9 trials is \( \left(\frac{31}{36}\right)^9 \). Therefore, the probability that 6 appears at least once, \( P_1 \), is:
\( P_1 = 1 - \left(\frac{31}{36}\right)^9 \).
Similarly, the probability that a sum of 8 does not appear in a single trial is the same, \( \bar{P_2} = \frac{31}{36} \). The probability that 8 does not appear in all 9 trials is \( \left(\frac{31}{36}\right)^9 \). Thus, the probability that 8 appears at least once, \( P_2 \), is:
\( P_2 = 1 - \left(\frac{31}{36}\right)^9 \).
Since \( P_1 \) and \( P_2 \) can be expressed similarly, \( P_1 : P_2 = 1 : 1 \).
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
Top AP EAPCET Probability Questions
S is the sample space and A, B are two events of a random experiment. Match the items of List A with the items of List B.
Then the correct match is:- If \( P(A \cap B) + P(B \mid A \cap B) = \), then:
- Two digits are selected at random from the digits 1 through 9. If their sum is even, then the probability that both are odd is:
For the probability distribution of a discrete random variable \( X \) as given below, the mean of \( X \) is:
- A shopkeeper buys a particular type of electric bulbs from three manufacturers $ M_1, M_2, M_3 $. He buys 25% of his requirement from $ M_1 $, 45% from $ M_2 $, and 30% from $ M_3 $. Based on past experience, he found that 2% of type $ M_1 $ bulbs are defective, whereas only 1% of type $ M_1 $ and type $ M_2 $ are defective. If a bulb chosen by him at random is defective, then the probability that it was of type $ M_3 $ is:
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is