Complete notes on experimental probability with examples and formula sheet
Probability is the branch of mathematics that deals with the likelihood or chance of an event occurring. It helps us measure uncertainty and make predictions about future outcomes based on past observations.
Probability is a measure of the chance that an event will occur. It is expressed as a number between 0 and 1.
If P(E) = 0, the event is impossible.
If P(E) = 1, the event is certain to happen.
Experiment: Tossing a coin
Possible Outcomes: Head (H) or Tail (T)
Sample Space: S = {H, T}
Event: Getting a Head = {H}
| Type of Event | Description | Example |
|---|---|---|
| Certain Event | Event that is sure to happen | Getting a number ā¤ 6 when rolling a die |
| Impossible Event | Event that cannot happen | Getting 7 when rolling a standard die |
| Equally Likely Events | Events with same chance of occurring | Getting Head or Tail in a fair coin toss |
| Complementary Events | Events where one must occur | Getting even or odd number on a die |
Experimental probability is based on actual experiments and observations. It is calculated from the results of performing an experiment many times.
Formula:
P(E) = Number of trials where event E occurred / Total number of trials
ā¢ It is based on actual observations from experiments.
ā¢ As the number of trials increases, experimental probability approaches theoretical probability.
ā¢ Different people may get different values for the same experiment.
ā¢ It gives more accurate results with a larger number of trials.
Q: A coin was tossed 100 times. Head appeared 56 times. Find the probability of getting a head.
Solution:
Number of trials = 100
Number of times head appeared = 56
P(Head) = 56/100 = 0.56 or 56%
Therefore, P(Head) = 0.56
Q: A die was rolled 200 times. The number 6 appeared 28 times. What is the experimental probability of getting 6?
Solution:
Total trials = 200
Number of times 6 appeared = 28
P(getting 6) = 28/200 = 7/50 = 0.14
Therefore, P(6) = 0.14 or 14%
If E is an event, then E' (or Ä) is its complement - the event that E does not occur.
P(E) + P(E') = 1
P(E') = 1 - P(E)
Q: If the probability of raining today is 0.7, what is the probability of not raining?
Solution:
P(Rain) = 0.7
P(No Rain) = 1 - P(Rain) = 1 - 0.7 = 0.3
Therefore, P(No Rain) = 0.3 or 30%
The sum of probabilities of all possible outcomes of an experiment is always equal to 1.
P(Eā) + P(Eā) + P(Eā) + ... + P(Eā) = 1
Q: In a bag, the probability of drawing a red ball is 0.3, blue ball is 0.25, and green ball is 0.2. Find the probability of drawing a ball of other color.
Solution:
P(Red) + P(Blue) + P(Green) + P(Other) = 1
0.3 + 0.25 + 0.2 + P(Other) = 1
0.75 + P(Other) = 1
P(Other) = 1 - 0.75 = 0.25
Therefore, P(Other color) = 0.25
Q: A factory produced 500 bulbs. On testing, 20 bulbs were found defective. What is the probability that a randomly selected bulb is defective?
Solution:
Total bulbs = 500
Defective bulbs = 20
P(Defective) = 20/500 = 1/25 = 0.04
Therefore, P(Defective bulb) = 0.04 or 4%
Q: In a survey of 1000 families, the following data was collected about the number of children:
0 children: 200 families, 1 child: 350 families, 2 children: 300 families, 3+ children: 150 families
Find the probability that a randomly chosen family has 2 children.
Solution:
Total families = 1000
Families with 2 children = 300
P(2 children) = 300/1000 = 0.3
Therefore, P(2 children) = 0.3 or 30%
P(E) = Favorable outcomes / Total trials
Based on actual experiments
0 ā¤ P(E) ā¤ 1
P(Impossible) = 0
P(Certain) = 1
Fundamental limits
P(E) + P(E') = 1
P(E') = 1 - P(E)
Sum equals 1
P(Eā) + P(Eā) + ... + P(Eā) = 1
All outcomes sum to 1
ā¢ Always identify the total number of trials first.
ā¢ Count the favorable outcomes carefully.
ā¢ Express probability as a fraction, decimal, or percentage.
ā¢ Check if your answer lies between 0 and 1.
ā¢ Use complementary probability when direct calculation is complex.