CHAPTER 3. COUNTING AND PROBABILITY O 102 „5 10 11 12 Figure 3.1: Sample space for rolling of two dice the sum principle we get that the probability of the sum being a multiple of 2 3 is 36 4 12 36 36 36 36 We conclude this chapter with an example similar to the last one. Only this time we have doctored dice. Example 3.2.10. Two distinct doctored dice, similar to the one on Exam- ple 3.2.2, page 93, are rolled and the two numbers on top are observed. What is the probability that the sum is a multiple of 5? Since the sum is a number between 2 and 12, we can break down the event E "sum is multiple of 5" into 2 disjoint cases "sum = 5" and "sum Thus pr(E) = pr(sum = 5) + pr(sum = 10). To determine the probabilities on the right side, we turn again to Figure 3.1 and this time to Table 3.2, 10". %3D page 94. Thus pr (sum = 5) = 0.1 x 0.35 + 0.1 x 0.1+0.1 × 0.1+0.35 × 0.1 0.09, 28. Consider the random experiment "Roll two identical dice and com- pute/record the difference". Find the sample space. Assuming the dice are fair, find the probability space. (Hint: use the diagonals in Figure 3.1, on page 102.) 29. Two distinguishable dice are rolled and the numbers on top are ob- served. Consider the event E: "one of the numbers is twice the other". a) Write E as as set of outcomes. b) If the the two dice are fair, what is the probability of E?
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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