How Can Teachers Build Notions of Conditional Probability and Independence

Research offers an emerging description of students' thinking in conditional probability and independence. Each of these concepts is associated with precise mathematical definitions that convey their interrelatedness. With respect to conditional probability, Hogg and Tanis (1993) point out that in some random experiments there is interest only in those outcomes that are elements of a subset B of the sample space S. Under these circumstances, the conditional probability of an event A given that event B has occurred, P(AIB), is the probability of A considering as possible outcomes only those outcomes of the random experiment that are elements of B. That is, the probability of event A is evaluated under the conditions of a new sample space, one that has been conditioned by the occurrence of event B. Hogg and Tanis also note that a special case of conditional probability occurs in a random experiment carried out in without-replacement situations. For example, consider an experiment where a gumball is selected and not replaced from a machine containing one red, one green, and one yellow gumball. The sample space immediately prior to the second draw will be a subset of the original sample space. The probability of "green," for example, will be conditional on the outcome of the first draw. If a green gumball is picked on the first draw, the probability of "green" given the event "green" on the first draw will be 0. On the other hand, if a red gumball is selected on the first draw, the probability of "green" given the event "red" on the first draw will be 0.5.

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