The probability that a randomly chosen 5 -digit number is made from exactly two digits is :
- $\frac{121}{10^{4}}$
- $\frac{150}{10^{4}}$
- $\frac{135}{10^{4}}$
- $\frac{134}{10^{4}}$
The Correct Option is C
Solution and Explanation
First Case: Choose two non-zero digits \({ }^{9} C _{2}\)
Now, number of 5 -digit numbers containing both digits \(=2^{5}-2\)
Second Case: Choose one non-zero & one zero as digit \({ }^{9} C _{1}\)
Number of 5 -digit numbers containing one non zero and one zero both \(=\left(2^{4}-1\right)\)
Required prob
\(=\frac{\left({ }^{9} C _{2} \times\left(2^{5}-2\right)+{ }^{9} C _{1} \times\left(2^{4}-1\right)\right)}{9 \times 10^{4}}\)
\(=\frac{36 \times(32-2)+9 \times(16-1)}{9 \times 10^{4}}\)
\(=\frac{4 \times 30+15}{10^{4}}=\frac{135}{10^{4}}\)
\(\text{Hence, the correct option is (C): }\) \(\frac{135}{10^{4}}\)
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Concepts Used:
Random Experiments
An Experiment is the activity that produces a result or an outcome. It is an element of uncertainty as to which it occurs when we perform an activity or experiment. Normally we get a different number of outcomes from an experiment. However, when an experiment satisfies the following two conditions, it is known as random experiment.
- It has more than one possible outcome.
- It is not possible to predict the outcome in advance.
On the basis of random experiment we can identify whether the given experiment is random or not. Let’s check with the help of example which is a random experiment and which is not.
Question: Using a calculator, divide 36 by 4. Now check, whether it is a random experiment or not.
Solution:
- This activity can be repeated under identical conditions though it has only one possible result.
- The outcome is always 9, which means we can predict the outcome each time we repeat the operation.
The given activity is not a random experiment.