If an unbiased die, marked with $-2,-1,0,1,2,3$ on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is:
- $\frac{440}{2592}$
- $\frac{881}{2592}$
- $\frac{27}{288}$
- $\frac{521}{2592}$
The Correct Option is D
Approach Solution - 1
We are given a die marked with the numbers \(-2, -1, 0, 1, 2, 3\). We need to find the probability that the product of the outcomes is positive when the die is thrown five times.
Step 1: Conditions for a Positive Product
The product of the outcomes will be positive if either:
- All outcomes are positive, or
- Any two outcomes are negative (since the product of two negative numbers is positive).
Step 2: Probabilities of Outcomes
Let \(p\) represent the probability of a positive outcome and \(q\) represent the probability of a negative outcome. The probabilities are calculated as follows:
\[ p = \frac{3}{6} = \frac{1}{2}, \quad q = \frac{2}{6} = \frac{1}{3}. \]
Step 3: Required Probability
The required probability can be calculated by considering the following cases:
- All 5 outcomes are positive.
- Any 2 outcomes are negative, and the rest are positive.
The total probability is given by:
\[ P(\text{positive}) = \binom{5}{5} \left(\frac{1}{2}\right)^5 + \binom{5}{2} \left(\frac{1}{3}\right)^2 \left(\frac{1}{2}\right)^3. \]
Step 4: Simplifying the Terms
Expand each term in the equation:
\[ P(\text{positive}) = \binom{5}{5} \left(\frac{1}{2}\right)^5 + \binom{5}{2} \left(\frac{1}{3}\right)^2 \left(\frac{1}{2}\right)^3. \]
Calculate the binomial coefficients and probabilities:
- \(\binom{5}{5} = 1\), so the first term is: \[ 1 \cdot \left(\frac{1}{2}\right)^5 = \frac{1}{32}. \]
- \(\binom{5}{2} = 10\), so the second term is: \[ 10 \cdot \left(\frac{1}{3}\right)^2 \cdot \left(\frac{1}{2}\right)^3 = 10 \cdot \frac{1}{9} \cdot \frac{1}{8} = \frac{10}{72} = \frac{5}{36}. \]
Combine the terms:
\[ P(\text{positive}) = \frac{1}{32} + \frac{5}{36}. \]
Step 5: Final Simplification
Find a common denominator and simplify:
\[ P(\text{positive}) = \frac{45}{2592} + \frac{521}{2592} = \frac{521}{2592}. \]
Final Answer
Thus, the correct probability is:
\[ \boxed{\frac{521}{2592}}. \]
Approach Solution -2
Required probability
Top Questions on Probability
- If a random variable \( X \) has the probability distribution:
\[ P(X = x) = \begin{cases} k, & \text{if } x = 0 \\ 2k, & \text{if } x = 1 \text{ or } 2 \\ 0, & \text{otherwise} \end{cases} \] then the value of \( k \) is - Two numbers are selected at random (without replacement) from the first six positive integers. Let $X$ denote the smaller of the two numbers obtained. Calculate the mathematical expectation of $X$.
- The mortality rate for a certain disease is 0.007. Using Poisson distribution, calculate the probability for 2 deaths in a group of 400 people. [Use $e^{-2.8} = 0.0608$]
- A die was thrown \( n \) times until the lowest number on the die appeared. If the mean is \( \frac{n}{g} \), then what is the value of \( n \)?
- MHT CET - 2025
- Mathematics
- Probability
- A bag contains 5 red, 3 blue, and 2 green balls. If two balls are drawn at random without replacement, what is the probability that both are red?
- BITSAT - 2025
- Mathematics
- Probability
Questions Asked in JEE Main exam
- The number of 6-letter words, with or without meaning, that can be formed using the letters of the word "MATHS" such that any letter that appears in the word must appear at least twice is:
- JEE Main - 2025
- Calculus
- Number of functions \( f: \{1, 2, \dots, 100\} \to \{0, 1\} \), that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
- JEE Main - 2025
- Vectors
- Let \( P_n = \alpha^n + \beta^n \), \( P_{10} = 123 \), \( P_9 = 76 \), \( P_8 = 47 \), and \( P_1 = 1 \). The quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:
- JEE Main - 2025
- Sequences and Series
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
- If \[ f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx, \quad f(0) = -6, { then } f(1) { is equal to:} \]
- JEE Main - 2025
- Calculus
Concepts Used:
Probability
Probability is defined as the extent to which an event is likely to happen. It is measured by the ratio of the favorable outcome to the total number of possible outcomes.
The definitions of some important terms related to probability are given below:
Sample space
The set of possible results or outcomes in a trial is referred to as the sample space. For instance, when we flip a coin, the possible outcomes are heads or tails. On the other hand, when we roll a single die, the possible outcomes are 1, 2, 3, 4, 5, 6.
Sample point
In a sample space, a sample point is one of the possible results. For instance, when using a deck of cards, as an outcome, a sample point would be the ace of spades or the queen of hearts.
Experiment
When the results of a series of actions are always uncertain, this is referred to as a trial or an experiment. For Instance, choosing a card from a deck, tossing a coin, or rolling a die, the results are uncertain.
Event
An event is a single outcome that happens as a result of a trial or experiment. For instance, getting a three on a die or an eight of clubs when selecting a card from a deck are happenings of certain events.
Outcome
A possible outcome of a trial or experiment is referred to as a result of an outcome. For instance, tossing a coin could result in heads or tails. Here the possible outcomes are heads or tails. While the possible outcomes of dice thrown are 1, 2, 3, 4, 5, or 6.