Question

A bag contains 5 black balls, and 7 green balls. Two balls are drawn simultaneously at random, what is the probability that both are different in colour? (1) \(17 / 33\) (2) \(35 / 132\) (3) \(5 / 66\) (4) \(35 / 66\)

Short answer

Answer: The probability of drawing two balls of different colors from the given bag is $\frac{35}{66}$.

Step-by-step solution

Find the total number of possibilities

For drawing two balls from a bag with 12 balls, we can use the combination formula: \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of balls and \(r\) is the number of balls we want to draw. Here, \(n=12\) and \(r=2\), so \(\binom{12}{2} = \frac{12!}{2!(10)!} = \frac{12 \times 11}{2 \times 1} = 66\). There are 66 possible combinations for drawing two balls from the bag.

Calculate the favorable outcomes

We need both balls to be different in color. Therefore, the desired combinations are: 1. One black ball and one green ball. We can draw 1 black ball from the bag containing 5 black balls in \(\binom{5}{1}\) ways, which is \(\frac{5!}{1!(4)!} = \frac{5 \times 4!}{1 \times 4!} = 5\) ways. Similarly, we can draw 1 green ball from the bag containing 7 green balls in \(\binom{7}{1}\) ways, which is \(\frac{7!}{1!(6)!} = \frac{7 \times 6!}{1 \times 6!} = 7\) ways. The number of favorable outcomes is the product of the ways of drawing one black ball and the ways of drawing one green ball, i.e., \(5 \times 7 = 35\).

Calculate the probability of getting different colored balls

Finally, we will divide the number of favorable outcomes by the total number of possible outcomes to get the probability of both balls being different in color: \(P = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{35}{66}\) So, the probability that both drawn balls are different in color is \(\frac{35}{66}\). The correct option is (4) \(\frac{35}{66}\).

Key concepts

Combination Formula

When it comes to combinatorial mathematics, the combination formula is an essential concept that allows us to figure out the number of ways we can select items from a group, where the order of selection does not matter. It's represented by the symbol \(\binom{n}{r}\), which reads as 'n choose r'. The formula is given by \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\], where \(n!\) (n factorial) is the product of all positive integers up to \(n\), and \(r\) is the number of items being chosen.

For example, if you have a set of twelve unique balls and wish to find out in how many ways you can select any two of them, you'd use the combination formula. Applying the formula by plugging in 12 for \(n\) and 2 for \(r\), we get \(66\) different combinations. This is crucial for calculating probabilities in situations where we're concerned with combinations of outcomes rather than permutations, which takes into account the arrangement order.

Favourable Outcomes in Probability

In probability, identifying favourable outcomes is key to determining the likelihood of a particular event. Favourable outcomes are those specific results that we're interested in among all possible outcomes in a probability experiment. To calculate the probability of an event, you take the number of favourable outcomes and divide it by the total number of possible outcomes.

Let's look at our example where we have a bag with black and green balls. If we want to draw two balls of different colors, the favourable outcomes are all those pairs that consist of one black ball and one green ball. As we saw in the solution steps for the original problem, it's possible to draw one black ball in \(5\) ways and one green ball in \(7\) ways, which leads us to \(35\) favourable combinations. These are the outcomes that meet our condition, and they serve as the numerator in our probability calculation.

Mathematical Probability

Mathematical probability is a measure of the likelihood that an event will occur. It's expressed as a number between 0 and 1, inclusive, where 0 indicates impossibility and 1 indicates certainty. Probability can often be thought of as the long-run relative frequency of an event's occurrence.

To calculate a probability, we typically divide the number of favourable outcomes by the total number of possible outcomes. Returning to our problem with the balls, after calculating \(35\) favourable outcomes and knowing there are \(66\) combinations we can draw, the probability of drawing two balls of different colors is \(\frac{35}{66}\), or approximately 0.53. This represents a bit over a half chance of drawing one ball of each color from the bag. Understanding mathematical probability is fundamental for predicting outcomes in random processes and assessing risks and odds in everyday scenarios.