Let \(X\) be the set of all five-digit numbers formed using \(1,2,2,2,2,4,4,0\). for example \(22240\) is in \(X\) while \(02244\) and \(44422\) are not in \(X\). Suppose that each element of \(X\) has an equal chance of being chosen. Let \(p\) be the conditional probability that a component chosen at random is a multiple of \(20 \) given that it is a multiple of \(5\). Then the numerous of \(38p\) is equal to
Correct Answer: 31
Solution and Explanation
Given: The task is to calculate the number of five-digit numbers divisible by 5, but not by 20. We are also given that the last digit of these numbers is either 0 or 5 (as numbers divisible by 5 must end in 0 or 5).
Step 1: Numbers Divisible by 5
The last digit of five-digit numbers divisible by 5 must be either 0 or 5. Let's check the numbers for each case:
For five-digit numbers ending in 0:
- 2224 → 4
- 2244 → 6
- 2221 → 4
- 2241 → 12
- 2441 → 12
The number of five-digit numbers divisible by 5 is 38.
Step 2: Numbers Divisible by 5 but Not by 20
To count numbers divisible by 5 but not by 20, we calculate numbers divisible by 5 but exclude those divisible by 20. Here is the set-up for finding numbers divisible by 20 but not by 5:
We are given that the total number of numbers divisible by 5 is 38, and the number of numbers divisible by both 5 and 20 is 7:
\(p = \frac{31}{38}\)
Step 3: Final Answer
After calculating, we find that the number of five-digit numbers divisible by 5 but not 20 is:
\(38p = 31\)
Therefore, the number of such numbers is 31.
Top Questions on 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
A shop selling electronic items sells smartphones of only three reputed companies A, B, and C because chances of their manufacturing a defective smartphone are only 5%, 4%, and 2% respectively. In his inventory, he has 25% smartphones from company A, 35% smartphones from company B, and 40% smartphones from company C.
(ii) What is the probability that this defective smartphone was manufactured by company B?
A person buys a smartphone from this shopA shop selling electronic items sells smartphones of only three reputed companies A, B, and C because chances of their manufacturing a defective smartphone are only 5%, 4%, and 2% respectively. In his inventory, he has 25% smartphones from company A, 35% smartphones from company B, and 40% smartphones from company C.
A person buys a smartphone from this shop(i) Find the probability that it was defective.
- In a consignment of electric lamps, 5% are defective. If a random sample of 8 lamps are inspected, what is the probability that one or more lamps are defective?
- When a coin is tossed 4 times, what is the probability of getting at most 3 heads?
Questions Asked in JEE Advanced exam
- A single slit diffraction experiment is performed to determine the slit width using the equation, $ \dfrac{b d}{D} = m \lambda $, where $ b $ is the slit width, $ D $ the shortest distance between the slit and the screen, $ d $ the distance between the $ m^{\text{th}} $ diffraction maximum and the central maximum, and $ \lambda $ is the wavelength. $ D $ and $ d $ are measured with scales of least count of 1 cm and 1 mm, respectively. The values of $ \lambda $ and $ m $ are known precisely to be $ 600 \, \text{nm} $ and 3, respectively. The absolute error (in $ \mu\text{m} $) in the value of $ b $ estimated using the diffraction maximum that occurs for $ m = 3 $ with $ d = 5 \, \text{mm} $ and $ D = 1 \, \text{m} $ is ___.
- JEE Advanced - 2025
- Error Propagation
- Let $$ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \cdots + \frac{1}{\sin 118^\circ \sin 119^\circ}. $$ Then the value of $$ \left( \frac{\csc 1^\circ}{\alpha} \right)^2 $$ is \rule{1cm}{0.15mm}.
- JEE Advanced - 2025
- Some Applications of Trigonometry
Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter $ D $ of a tube. The measured value of $ D $ is:
- JEE Advanced - 2025
- Units and measurement
- Let $ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $ and $ P = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix} $. Let $ Q = \begin{pmatrix} x & y \\ z & 4 \end{pmatrix} $ for some non-zero real numbers $ x, y, z $, for which there is a $ 2 \times 2 $ matrix $ R $ with all entries being non-zero real numbers, such that $$ QR = RP $$ Then which of the following statements is (are) TRUE?
- JEE Advanced - 2025
- Matrices
- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
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.