The problem asks for the probability that the product of two numbers, one drawn from each box, is an even number.
Analyzing Number Properties
First, let's list the numbers in each box and identify whether they are odd or even.
| Box | Numbers | Odd Numbers | Even Numbers | Count (Odd) | Count (Even) | Total Count |
|---|---|---|---|---|---|---|
| 1 | $3, 6, 9, 12, 15$ | $3, 9, 15$ | $6, 12$ | 3 | 2 | 5 |
| 2 | $6, 11, 16, 21, 26$ | $11, 21$ | $6, 16, 26$ | 2 | 3 | 5 |
Calculating Total Outcomes
The total number of possible outcomes is the product of the number of chips in each box.
Total Outcomes = (Chips in Box 1) $\times$ (Chips in Box 2) = $5 \times 5 = 25$.
Probability of an Odd Product
The product of two numbers is odd only if both numbers are odd. It's often easier to calculate the probability of the complementary event (product being odd) and subtract it from 1.
- The probability of drawing an odd number from Box 1 is $P(\text{Odd}_1) = \frac{\text{Number of odd chips in Box 1}}{\text{Total chips in Box 1}} = \frac{3}{5}$.
- The probability of drawing an odd number from Box 2 is $P(\text{Odd}_2) = \frac{\text{Number of odd chips in Box 2}}{\text{Total chips in Box 2}} = \frac{2}{5}$.
- The probability that both drawn numbers are odd (resulting in an odd product) is: $P(\text{Odd Product}) = P(\text{Odd}_1) \times P(\text{Odd}_2) = \frac{3}{5} \times \frac{2}{5} = \frac{6}{25}$.
Probability of an Even Product
The probability of the product being even is 1 minus the probability of the product being odd.
$P(\text{Even Product}) = 1 - P(\text{Odd Product})$
$P(\text{Even Product}) = 1 - \frac{6}{25} = \frac{25}{25} - \frac{6}{25} = \frac{19}{25}$.
Conclusion
The probability for the product to be an even number is $\frac{19}{25}$.
Important Questions from Basics of Probability
Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ______ . (rounded off to three decimal places)
- A quadratic polynomial $(x - \alpha)(x - \beta)$ over complex numbers is said to be square invariant if $(x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random.
The probability that the roots of the chosen polynomial are equal is ____________ (rounded off to one decimal place) - If, out of 50 fish of a pond, 12 had no ectoparasite and remaining had varying numbers of parasites on them, then how many of fish had at least one parasite ?
For a distribution with mean, median, mode and standard deviation 25, 24, 26 and 5 respectively, Karl Pearson's coefficient of skewness equals to:
Suppose two distributions have the same mean, same standard deviation and are equally skewed, but the first distribution is more peaked than the
other. This feature of the frequency distributions in which they are different are called:
