Question

A jar contains 10 blue marbles, 7 red marbles, and 4 yellow marbles. Without looking, you reach into the jar and choose one marble. What is the probability that the marble is red?

Short answer

The probability of drawing a red marble from the jar is 1/3.

Step-by-step solution

Identifying the Total Outcomes

There are a total of \(10 + 7 + 4 = 21\) marbles in the jar, so we have 21 total outcomes. This is because each marble represents a possible outcome when one is drawn from the jar.

Identifying the Favourable Outcomes

We are trying to find the probability of drawing a red marble, the number of favourable outcomes is given by the number of red marbles in the jar, which is 7.

Calculating the Probability

The probability \(P\) of an event is calculated by dividing the number of favourable outcomes \(F\) by the total number of outcomes \(T\). Using this formula, the probability \(P\) of drawing a red marble is \(P = F/T = 7/21\). This simplifies to \(P = 1/3\).

Key concepts

Calculating Probabilities

When we're tasked with calculating probabilities, our primary goal is to determine the likelihood, or chance, of a specific event occurring. Probability is fundamentally a ratio where the numerator (top number) represents the number of ways an event can happen (favorable outcomes), and the denominator (bottom number) denotes the total number of possible outcomes.

For instance, considering our jar with marbles, to calculate the probability of selecting a red marble, we first need to understand that each marble represents a unique potential outcome. Once we have the count of both the favorable and possible outcomes, we simply divide them. Thus, in mathematical terms:
\[\begin{equation}P = \frac{F}{T}\text{where}P = \text{Probability of the event},F = \text{Number of favorable outcomes},T = \text{Total number of outcomes}.\text{\end{equation}\]}
It's crucial to simplify the fraction to its lowest terms to convey the probability in the simplest form, just as we did with the marbles, arriving at \( P = \frac{1}{3} \).

Favorable Outcomes

Key to grasping probability is understanding what is meant by favorable outcomes. These are simply the outcomes that align with the specific event we're interested in. It's important to count only these outcomes when assessing the probability of an event.

For example, if we're interested in drawing a red marble from our jar, we focus exclusively on the red marbles. The number of red marbles - 7 in this case - represents the favorable outcomes. This count of favorable outcomes serves as our numerator in the probability ratio. Remember, if there are multiple ways an event can occur, each one needs to be counted as a favorable outcome.

Total Outcomes

Conversely, total outcomes encompass all possible scenarios that could occur when an event takes place, regardless of the color or characteristic of the outcome. It's the big picture - the complete set of events we could see from our action (such as drawing a marble from a jar).

In our marble example, the total outcomes would be the sum of all the marbles, since any one of them could have been drawn. By adding up the 10 blue, 7 red, and 4 yellow marbles, we see there are 21 total outcomes. It's vital to ensure our count of possible outcomes is exhaustive as this number denominates our probability fraction, providing the context needed to understand what the favorable outcomes mean.