Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates an unlikely event, 1 indicates a certain event, and values in between represent varying degrees of likelihood.
Probability is used in many real-world scenarios, such as weather forecasting, sports predictions, and games of chance.
Experimental probability is estimated from observed outcomes. Example: flipping a coin 100 times, heads appear 60 times → experimental probability of heads = 60/100 = 0.6.
Theoretical probability is based on mathematical calculation assuming all outcomes are equally likely. Example: rolling a fair six-sided die, probability of 3 = 1/6 ≈ 0.167.
Approaches include:
Example: rolling a fair six-sided die 100 times and recording frequencies to develop a probability model.
Experimental probability approaches theoretical probability as the number of trials increases.
If experimental and theoretical probabilities are close, the model is accurate; discrepancies may indicate sampling error or bias.
Probability models help find probabilities of events and can be used with graphical displays or numerical summaries to make informal inferences about populations or samples.
Fair coin tossed 50 times: 30 heads, 20 tails
Step 1: Experimental Probability
Heads = 30/50 = 0.6
Tails = 20/50 = 0.4
Step 2: Theoretical Probability
Heads = 0.5, Tails = 0.5
Step 3: Interpretation
Experimental probabilities close to theoretical values; slight differences due to sample size or chance
Roll six-sided die 100 times:
Frequencies:
1:15, 2:20, 3:25, 4:10, 5:20, 6:10
Step 1: Probability Model
P(1)=0.15, P(2)=0.2, P(3)=0.25, P(4)=0.1, P(5)=0.2, P(6)=0.1
Step 2: Compare to Theoretical Probability
Each face of fair die = 1/6 ≈ 0.167
Step 3: Interpretation
Use model to estimate likelihood of each outcome and compare with theoretical probability for accuracy