3.1.4 Addition and multiplication rules Addition rule Section 3.1.1 noted that, for a discrete distribution, the probability that the next measured value of x is either x; or x; equals P(x) + P(x;), unless i = j. The key point is that x can't equal both x; and x;; we say that the alternative values are exclusive. More generally, the probability that a person is either taller than 2 m or shorter than 1.9 m is obtained by addition, whereas the probability of being either taller than 2 m or nearsighted cannot be obtained in this way. For a continuous distribution, the probability that the next measured value of x is either between a and b or between c and d equals the sum, fodx P(x) + dx P(x), provided the two intervals don't overlap. This result follows because the two proba- bilities (to be between a and b or between c and d) are exclusive in this case. Multiplication rule Now suppose that we measure two independent quantities, for example, tossing a coin and rolling a die. What is the probability that we get heads and roll a 6? To find out, just list all 2 x 6 = 12 possibilities. Each is equally probable, so the chance of getting the specified outcome is. This example shows that the joint probability distribution for two independent events is the product of the two simpler distributions. Let Pjoint (xi, yk) be the joint distribution, where i = 1 or 2 and x₁ =(heads), x2 =(tails); similarly, yк = K, the number on the die. Then the multiplication rule says Your Turn 3C Pjoint (xi, yk) = Pcoin (xi) × Pdie (YK). == (3.15) Equation 3.15 is correct even for loaded dice (the Pdie (yk) aren't all equal to }) or a two-headed coin (Pcoin (x1) = 1, Pcoin (X2) === 0). On the other hand, for two connected events (for example, the chance of rain versus the chance of hail), we don't get such a simple relation. Show that if Pcoin and Pdie are correctly normalized, then so will be Pjoint.