Q9(i):
A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (i) red ?

Solution :

Given:

The number of red marbles in the box = $5$

The number of white marbles in the box = $8$

The number of green marbles in the box = $4$

To Find:

The probability that a marble taken out at random is red.

Visual Representation of the Sample Space:

Step 1: Calculate the total number of possible outcomes.

Let $n(S)$ be the total number of marbles in the box. Since the marble is drawn at random, every marble has an equal chance of being selected.

$n(S) = \text{Number of red marbles} + \text{Number of white marbles} + \text{Number of green marbles}$

$n(S) = 5 + 8 + 4$

$n(S) = 17$

[Since the total number of elementary events in the sample space is the sum of all individual counts of marbles.]

Step 2: Identify the number of favorable outcomes.

Let $E$ be the event of drawing a red marble.

Let $n(E)$ be the number of favorable outcomes for event $E$.

$n(E) = \text{Number of red marbles} = 5$

Step 3: Apply the Probability Formula.

The probability of an event $P(E)$ is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.

$P(E) = \frac{n(E)}{n(S)}$

[Using the classical definition of probability for equally likely outcomes.]

Step 4: Substitution and Calculation.

$P(\text{Red}) = \frac{5}{17}$

Since $5$ and $17$ are coprime numbers (where $17$ is a prime number and not a factor of $5$), the fraction cannot be simplified further.

Final Answer: The probability that the marble taken out will be red is $\frac{5}{17}$.

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