Two-Way Tables

Key Questions

• How do you calculate marginal, joint, and conditional probabilities from a two-way table?

  If you are given a pmf = `p_(XY)(x,y)`

  and you would like to find the marginal `p_Y(y)`

  we would use the formula `p_y(y) = sum_ip(x_i,y)`

  in other words you would sum over all of `x` at the point `y`

  So if we look at this table and want to find the marginal `p_Y(3)`

  we go:

  `p_Y(3) = P( Y = 3)`
  `= P(Y = 3, X = 3) + P(Y = 3, X=4)`
  `= 0.1 + 0.2`
  `=0.3`

  Now to look at the formula for the conditional probability

  we can look at the formula for `x` given `y` which is a conditional probability.

  `p_(X|Y)(x|y) = P(X = x_i | Y = y_j) = (P(X = x_i, Y= y_j))/(P(Y = y_j))`

  `=(p_(XY)(x_i,y_j))/(p_Y(y_i))`

  now to use an example, we will look back at our table.

  let us look for the conditional probability of:

  `p_(X|Y)(3|4) = 0.1/0.4 = 0.25`

  Thus, the probability that `X = 3` given that `Y=4` is `0.25`

  Miles A. · 1 · May 18 2015
• What is a two-way table?

  A two-way table is a display of data divided into two different categories of subsets.

  In the example below, the categories are
  age range: with subsets for ages 0-5, 6-10, and 11-15
  color preference: with subsets for various color choices

  The entry at D5 (value 8) indicates that 8 children in the age range 6-10 chose yellow as their preferred color.

  The sum of values across a line indicates the number of children who chose the color for that line across all age ranges.

  The sum of values down a column indicates the number of children surveyed in the corresponding age range.

  Alan P. · 1 · Feb 17 2015

• Consider a two-way table that summarizes age and ice cream preference. Of the children, 7 prefer vanilla, 12 prefer chocolate, and 3 prefer butter pecan. Of the adults, 10 prefer vanilla, 31 prefer chocolate, and 8 prefer butter pecan. What is the joint probability that a randomly selected person is a child who prefers chocolate?
• What is a joint probability in a two-way table?
• What is a conditional probability in a two-way table?
• Consider a two-way table that summarizes age and ice cream preference. Of the children, 7 prefer vanilla, 12 prefer chocolate, and 3 prefer butter pecan. Of the adults, 10 prefer vanilla, 31 prefer chocolate, and 8 prefer butter pecan. What is the conditional probability that a person prefers vanilla, GIVEN THAT the person is an adult?
• Consider a two-way table that summarizes age and ice cream preference. Of the children, 7 prefer vanilla, 12 prefer chocolate, and 3 prefer butter pecan. Of the adults, 10 prefer vanilla, 31 prefer chocolate, and 8 prefer butter pecan. What is the probability that a randomly selected individual is a child, GIVEN THAT the person prefers butter pecan?
• Consider a two-way table that summarizes age and ice cream preference. Of the children, 7 prefer vanilla, 12 prefer chocolate, and 3 prefer butter pecan. Of the adults, 10 prefer vanilla, 31 prefer chocolate, and 8 prefer butter pecan. Are the events "chocolate" and "child" independent? Why or why not?
• What is a two-way table?
• How do you calculate marginal, joint, and conditional probabilities from a two-way table?
• Question #b0d38
• How can we divide the salary data `{199,200,202,203,207,208,208,209,210,210,210,210,212,213,214,215,217,218,218,221}` in six classes?
• How does probability relate to the punnett square and the offspring?