Question

Do you multiply to find probability?

Answer 1

Hint: To find probability, either addition or multiplication is used. The choice of the operation depends on the event or events that are occurring. When the events that occur consecutively are independent, then we multiply to find the total probability. And when the events can either be one or the another, then in that case we add to find the probability.

Complete step by step solution:
According to the given question, we are asked when we multiply to find the probability. We will start from the scratch and learn when we multiply or add to find probability.
Probability of an event describes how likely that event is going to occur. It is represented by a numerical value that portrays the possibility of that event to occur. Probability of any event varies between 0 and 1, where a probability of 0 refers to an uncertain event and 1 refers to a certain event.
There are two type of events, that could occur:
1) Dependent events – When the occurrence of the first event influences the second event. Then, the events are dependent.
2) Independent events – When the events occur without being affected by the occurrence of other events, then, the events are called independent events.
So, in probability, there may be cases when we have to add or multiply to find the probability. Let’s understand these through an example.
Case 1: Finding probability using multiplication-
For example – Suppose we have a bag filled with 15 balls. Out of which 7 are red balls and 8 are yellow balls. Then what will be the probability of drawing a red ball first and then a yellow ball? And we are replacing the ball back into a bag after drawing each ball out.
Solution- We have a bag of 15 balls, with 7 red balls and 8 yellow balls. We have to draw a red ball first and then a yellow ball. Since, we have to replace the ball back into the bag, so the events are independent. If suppose, replacement was not there, then after drawing the first ball, drawing the second ball will get affected and then these will be dependent events.
Back to our question,
The probability of drawing a red ball first $$={\displaystyle \frac{7}{15}}=0.47$$
And then after replacement, probability of drawing a yellow ball $$={\displaystyle \frac{8}{15}}=0.53$$
So, the total probability $$={\displaystyle \frac{7}{15}}\times {\displaystyle \frac{8}{15}}={\displaystyle \frac{56}{225}}=0.249$$
Hence, we used multiplication to find the probability.

Case 2: Find probability using addition-
For example - Suppose we have a bag filled with 15 balls. Out of which 7 are red balls and 8 are yellow balls. Then what will be the probability of drawing a red ball first and then a yellow ball or a yellow ball first and then a red ball? And we are replacing the ball back into a bag after drawing each ball out.
Solution - We have a bag of 15 balls, with 7 red balls and 8 yellow balls. We have to draw a red ball first and then a yellow ball or a yellow ball first and then a yellow ball. Since, we have to replace the ball back into the bag, so the events are independent. But since at our first draw, we can have either red or yellow and then the second draw must be different from the first one. So we will be adding the two scenarios to get the total probability.
The probability of drawing a red ball first $$={\displaystyle \frac{7}{15}}=0.47$$
And then after replacement, probability of drawing a yellow ball $$={\displaystyle \frac{8}{15}}=0.53$$
Total probability = probability of red ball first and then a yellow ball + probability of yellow ball first then a red ball
$$={\displaystyle \frac{7}{15}}\times {\displaystyle \frac{8}{15}}+{\displaystyle \frac{8}{15}}\times {\displaystyle \frac{7}{15}}$$
$$={\displaystyle \frac{56}{225}}+{\displaystyle \frac{56}{225}}$$
$$={\displaystyle \frac{112}{225}}$$

Note: A rule of the thumb is, if in the question of probability, the word ‘and’ means that multiplication is required to find the probability (given that the consecutive events are independent). And when the ‘or’ is used, it would mean that we will require addition to find the probability.