11.5 Or and And Problems 707 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = = ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⋅ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⋅ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = = ⋅ ⋅ = P P P P first patient reacted favorably, the second patient reacted unfavorably, and the third patient was unaffected first patient reacted favorably second patient reacted unfavorably third patient was unaffected 19 25 2 24 4 23 19 1725 c) If none of the patients reacted favorably, the patients either reacted unfavorably or were unaffected. Six patients did not react favorably (2 reacted unfavorably and 4 were unaffected). The probability that the first patient selected did not react favorably is . 6 25 After the first patient is selected, 5 of the remaining 24 patients did not react favorably. After the second patient is selected, 4 of the remaining 23 patients did not react favorably. ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⋅ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⋅ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = ⋅ ⋅ = P P P P none reacted favorably first patient did not react favorably second patient did not react favorably third patient did not react favorably 6 25 5 24 4 23 1 115 d) In Section 11.4, we learned that P P (event happening at least once) 1 (event does not happen) = − In part (c), we determined that the probability of selecting three patients none of whom reacted favorably was . 1 115 Therefore, the probability that at least one of the patients selected reacted favorably can be determined as follows. ⎛ ⎝⎜ ⎞ ⎠⎟ = − ⎛ ⎝⎜ ⎞ ⎠⎟ = − = − = P P at least one of the three patients reacted favorably 1 none of the three patients reacted favorably 1 1 115 115 115 1 115 114 115 7 Now try Exercise 77 Timely Tip Which formula to use It is sometimes difficult to determine when to use the or formula and when to use the and formula. The following information may be helpful in deciding which formula to use. Or formula Or problems will almost always contain the word or in the statement of the problem. For example, determine the probability of drawing a heart or a 6. Or problems generally involve only one selection. For example, “one card is drawn” or “one die is rolled.” And formula And problems often do not use the word and in the statement of the problem. For example, “determine the probability that both cards drawn are red” or “determine the probability that none of those drawn is a banana” are both and -type problems. And problems will generally involve more than one selection. For example, the problem may read “two cards are drawn” or “three coins are tossed.” Exercises Warm Up Exercises In Exercises 1–10, fill in the blank with an appropriate word, phrase, or symbol(s). 1. Probability problems that contain the words and or or are considered ________ probability problems. Compound 2. Probability problems that require obtaining a successful outcome for at least one of the given events are ________ probability problems. Or 3. Probability problems that require obtaining a favorable outcome in each of the given events are ________ probability problems. And 4. If it is impossible for two events A and B to occur simultaneously, then the events are considered to be ________ exclusive. Mutually 5. For two events A and B, if the occurrence of either event in no way affects the probability of the occurrence of the other event, then the two events are considered to be ________ events. Independent 6. For two events A and B, if the occurrence of either event has an effect on the probability of the occurrence of the other event, then the two events are considered to be ________ events. Dependent 7. Experiments done with replacement will result in ________ events. Independent 8. Experiments done without replacement will result in ________ events. Dependent SECTION 11.5
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