One way to find the probability of an event is to conduct an experiment. Experimental probability of an event can defined as the ratio of the number of times the event occurs to the total number of trials. For example Sam rolled a number cube 50 times. A 3 appeared 10 times. Then the experimental probability of rolling a 3 is 10 out of 50 or 20%.
| Example
A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the experimental probability of getting a blue marble. Solution: · Take a marble from the bag. · Record the color and return the marble. · Repeat a few times (maybe 10 times). · Count the number of times a blue marble was picked (Suppose it is 6). The experimental probability of getting a blue marble from the bag is |
Experimental probability is frequently used in research and experiments of social sciences, behavioural sciences, economics and medicine.
In cases where the theoretical probability cannot be calculated, we need to rely on experimental probability.
For example, to find out how effective a given cure for a pathogen in mice is, we simply take a number of mice with the pathogen and inject our cure.
We then find out how many mice were cured and this would give us the experimental probability that a mouse is cured to be the ratio of number of mice cured to the total number of mice tested.
In this case, it is not possible to calculate the theoretical probability. We can then extend this experimental probability to all mice. It should be noted that in order for experimental probability to be meaningful in research, the sample size must be sufficiently large.
In our above example, if we test our cure on 3 mice and all of these are cured, then the experimental probability that a mouse is cured is 1. However, the sample size is too small to conclude that the cure works in 100% of the cases.
3.3 THEORETICAL PROBABILITY
Theoretical probability is the ratio between the number of ways an event can occur and the total number of possible outcomes in the sample space. Put simply, it’s the chance that something will happen, usually expressed as a percentage.
The formula for theoretical probability of an event is
| Example
A bag contains 20 red marbles, 8 blue marbles and 12 yellow marbles. Find the theoretical probability of getting a blue marble. There are 8 blue marbles. Therefore, the number of favourable outcomes = 8. There are a total of 20 marbles. Therefore, the number of total outcomes = 20 |
| Example
Find the probability of rolling an even number when you roll a die containing the numbers 1-6. Express the probability as a fraction, decimal, ratio and percent. Solution: The possible even numbers are 2, 4, 6. Number of favourable outcomes = 3. Total number of outcomes = 6 The probability = (fraction) = 0.5 (decimal) = 1:2 (ratio) = 50% (percent)
|
| Exercise
A coin is tossed 60 times. 27 times head appeared. Find the experimental probability of getting heads. |
Tree Diagram
A diagram used in strategic decision making, valuation or probability calculations. The diagram starts at a single node, with branches emanating to additional nodes, which represent mutually exclusive decisions or events. In the diagram below, the analysis will begin at the first blank node. A decision or event will then lead to node A or B. From these secondary nodes, additional decisions or events will occur leading to the third level of nodes, until a final conclusion is reached.
Using the diagram is simple once you assign the appropriate values to each node. Chance nodes, representing a possible outcome, must be assigned a probability. Decision nodes ask a question and must be followed by answer nodes, such as “yes” or “no”. Often, a value will be associated with a node, such as a cost or a payout. Tree diagrams combine the probabilities, decisions, costs and payouts of a decision and provide a strategic answer.
| Example:
A bag contains 3 black balls and 5 white balls. Paul picks a ball at random from the bag and replaces it back in the bag. He mixes the balls in the bag and then picks another ball at random from the bag. a) Construct a probability tree of the problem. b) Calculate the probability that Paul picks: i) two black balls ii) a black ball in his second draw Solution:
a) Check that the probabilities in the last column add up to 1.
b) i) To find the probability of getting two black balls, first locate the B branch and then follow the second B branch. Since these are independent events we can multiply the probability of each branch.
ii) There are two outcomes where the second ball can be black. Either (B, B) or (W, B)
From the probability tree diagram, we get: P(second ball black) = P(B, B) or P(W, B) = P(B, B) + P(W, B)
|
Leave a Reply
You must be logged in to post a comment.