multiplication rule

Examples of multiplication rule in the following topics:

• The Multiplication Rule

• The Multiplication Rule

  • The multiplication rule states that the probability that $A$ and $B$ both occur is equal to the probability that $B$ occurs times the conditional probability that $A$ occurs given that $B$ occurs.
  • This rule can be written:
  • We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator.
  • That is, in the equation $\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$, if we multiply both sides by $P(B)$, we obtain the Multiplication Rule.
  • Apply the multiplication rule to calculate the probability of both $A$ and $B$ occurring

• General multiplication rule

  • Section 2.1.6 introduced the Multiplication Rule for independent processes.
  • Here we provide the General Multiplication Rule for events that might not be independent.
  • This General Multiplication Rule is simply a rearrangement of the definition for conditional probability in Equation (2.40) on page 83.
  • We will compute our answer using the General Multiplication Rule and then verify it using Table 2.16.
  • Among the 96.08% of people who were not inoculated, 85.88% survived:P(result = lived and inoculated = no) = 0.8588 × 0.9608 = 0.8251 This is equivalent to the General Multiplication Rule.

• Student Learning Outcomes

• Summary of Formulas

• Independence

  • Examples 2.5 and 2.28 illustrate what is called the Multiplication Rule for independent processes.
  • Then we can compute whether a randomly selected person is right-handed and female using the Multiplication Rule:
  • We apply the Multiplication Rule for independent processes to determine the probability that both will be left-handed: 0.09 x 0.09 = 0.0081.
  • Since each are independent, we apply the Multiplication Rule for independent processes: P(all five are RH) = P( first = RH, second = RH, ..., fth = RH) = P( first = RH) x P(second = RH) x...x P( fth = RH) = 0.91 x 0.91 x 0.91 x 0.91 x 0.91 = 0.624 (b) Using the same reasoning as in (a), 0.09 x 0.09 x 0.09 x 0.09 x 0.09 = 0.0000059 (c) Use the complement, P(all five are RH), to answer this question: P(not all RH) = 1 - P(all RH) = 1 - 0:624 = 0:376

• Counting Rules and Techniques

  • Several useful combinatorial rules or combinatorial principles are commonly recognized and used.
  • The rule of sum (addition rule), rule of product (multiplication rule), and inclusion-exclusion principle are often used for enumerative purposes.
  • The rule of sum is an intuitive principle stating that if there are $a$ possible ways to do something, and $b$ possible ways to do another thing, and the two things can't both be done, then there are $a + b$ total possible ways to do one of the things.
  • The rule of product is another intuitive principle stating that if there are $a$ ways to do something and $b$ ways to do another thing, then there are $a \cdot b$ ways to do both things.
  • The inclusion-exclusion principle is a counting technique that is used to obtain the number of elements in a union of multiple sets.

• Fundamentals of Probability

  • The most basic and most important rules are listed below.
  • These events are called complementary events, and this rule is sometimes called the complement rule.
  • This is often called the multiplication rule.
  • We consider each of the five rules above in the context of this example.
  • Outline the most basic and most important rules in determining the probability of an event

• Multiple Regression Models

  • Multiple regression is used to find an equation that best predicts the $Y$ variable as a linear function of the multiple $X$ variables.
  • You use multiple regression when you have three or more measurement variables.
  • One use of multiple regression is prediction or estimation of an unknown $Y$ value corresponding to a set of $X$ values.
  • Multiple regression would give you an equation that would relate the tiger beetle density to a function of all the other variables.
  • As you are doing a multiple regression, there is also a null hypothesis for each $X$ variable, meaning that adding that $X$ variable to the multiple regression does not improve the fit of the multiple regression equation any more than expected by chance.

• Interpreting a Confidence Interval

  • For non-standard applications, there are several routes that might be taken to derive a rule for the construction of confidence intervals.
  • Established rules for standard procedures might be justified or explained via several of these routes.
  • Typically a rule for constructing confidence intervals is closely tied to a particular way of finding a point estimate of the quantity being considered.
  • A naive confidence interval for the true mean can be constructed centered on the sample mean with a width which is a multiple of the square root of the sample variance.
  • The confidence interval can be expressed in terms of samples (or repeated samples): "Were this procedure to be repeated on multiple samples, the calculated confidence interval (which would differ for each sample) would encompass the true population parameter 90% of the time. " Note that this does not refer to repeated measurement of the same sample, but repeated sampling.