Question

A bag contains 3 red, 5 blue and 7 green coloured balls. Find the probability of selecting a blue ball from the bag. (1) \(\frac{3}{15}\) (2) \(\frac{1}{3}\) (3) \(\frac{1}{4}\) (4) \(\frac{2}{3}\)

Short answer

Answer: The probability of selecting a blue ball is 1/3.

Step-by-step solution

Count the total number of balls in the bag

Add the number of red, blue, and green balls in the bag to find the total number of balls: $$ 3 + 5 + 7 = 15$$ So, there are 15 balls in the bag.

Identify the number of favorable outcomes

In this case, the favorable outcomes are the blue balls. There are 5 blue balls in the bag.

Calculate the probability

Apply the formula for probability using the values from steps 1 and 2: $$ P(E) = \frac{n(E)}{n(S)} = \frac{5}{15}$$

Simplify the probability

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (GCD), which in this case is 5: $$ P(E) = \frac{5}{15} = \frac{5 \div 5}{15 \div 5} = \frac{1}{3}$$

Choose the correct answer from the given options

The probability of selecting a blue ball is \(\frac{1}{3}\). The correct answer is option (2).

Key concepts

probability calculations

Understanding probability calculations is essential in mathematics when dealing with the likelihood of events. Probability is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The formula for calculating the probability of an event E happening is given by:

\[\begin{equation} P(E) = \frac{n(E)}{n(S)}\text{,}\tag{1}\end{equation}\]
where:

• \( n(E) \) is the number of favorable outcomes for event E,
• \( n(S) \) is the total number of possible outcomes in the sample space S.

When applying this formula, it often requires simplifying the resulting fraction to find the simplest form, which makes the probability easier to understand and compare with other probabilities. When simplifying, search for the greatest common divisor (GCD) of the numerator and the denominator, divide both by this number, and you'll get the simplified probability value.

favorable outcomes

Favorable outcomes are at the heart of probability calculations. In any given scenario, these are the outcomes that would lead to the event we are interested in. For instance, if you want to know the probability of drawing a blue ball from a bag, the favorable outcomes are simply all the instances of blue balls within the bag.

To calculate probability accurately, identifying favorable outcomes is crucial. They must be counted carefully because any mistake in this count directly affects the accuracy of the probability. One way to ensure no mistakes are made is by clearly defining what counts as a favorable outcome before you start counting, allowing for focused and effective counting without confusion. Finally, the counted favorable outcomes will serve as the numerator in the probability formula.

total number of outcomes

The total number of outcomes refers to all the possible results that could occur in a given situation, defined as the sample space S. When calculating probability, it is essential to consider every single possible outcome; omitting any can lead to incorrect probability calculations. The total number of outcomes includes all favorable and unfavorable outcomes combined.

In the context of our example with the colored balls, the total number is the sum of all the red, blue, and green balls. It's the denominator in the probability formula. To ensure accuracy, it's helpful to recount the total number of outcomes and confirm there weren't any mistakes. After establishing this number, you can proceed with the probability calculation formula, combining it with the count of favorable outcomes.