748 CHAPTER 11 Probability d) We are determining the probability of selecting exactly 3 red balls in 3 independent trials. Thus, x 3 = and n 3. = We determine P(3) as follows. P x C p q P C ( ) ( ) (3) ( ) 1 3 2 3 1 1 3 2 3 1 1 27 (1) 1 27 n x x n x 3 3 3 3 3 3 0 = = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ = ⎛ ⎝ ⎞ ⎠ = − − 7 Now try Exercise 13 All the probabilities obtained in Example 1 agree with the answers obtained by using the tree diagram. Whenever you obtain a value for P x( ), you should obtain a value between 0 and 1, inclusive. If you obtain a value greater than 1, you have made a mistake. Example 2 Rolling a Die A single die is rolled 5 times. a) Write the binomial probability formula used to determine the probability that x out of the 5 rolls will result in the number 4. b) Write the binomial probability formula used to determine the probability that 2 out of the 5 rolls will result in the number 4. Evaluate the formula using a scientific or graphing calculator. Round your answer to five decimal places. Solution a) The general binomial probability formula is P x C p q ( ) ( ) n x x n x = − Here, the number of trials of the experiment, n, is 5. A success is considered rolling a 4 on a single trial; therefore, the probability of success, p, is , 1 6 and the probability of failure, q, is . 5 6 Substituting these values into the general binomial probability formula, we have P x C ( ) ( ) 1 6 5 6 x x x 5 5 = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝⎜ ⎞ ⎠⎟ − b) We are asked to write the binomial probability formula used to determine the probability that 2 out of the 5 rolls will result in the number 4. We use the formula from part a) with x 2. = P C (2) ( ) 1 6 5 6 10 1 6 5 6 10 1 36 125 216 0.16075 5 2 2 5 2 2 3 = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛ ⎝ ⎞ ⎠ ⎛ ⎝⎜ ⎞ ⎠⎟ ≈ − Thus, the probability that 2 out of the 5 rolls will result in the number 4 is about 0.16075. 7 Now try Exercise 11 Applets Simulation Dice Rolling StatCrunch
RkJQdWJsaXNoZXIy NjM5ODQ=