A bag contains five purple ("P") blocks and one yellow ("Y") blocks. Without looking into the bag you reach in, take out a block, return the block to the bag, and then repeat the same process a number of times. Which of the following sequences of selected blocks is more likely?
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To determine which sequence of selected blocks is more likely, we need to analyze the probability of each sequence occurring.
Let's consider both options:
Option 1: Selecting two purple blocks followed by a yellow block (PPY)
To calculate the probability of this sequence, we need to consider the probability of selecting a purple block on each draw. Since there are five purple blocks and six total blocks in the bag, the probability of selecting a purple block on the first draw is 5/6. Since we return the block to the bag after each draw, the probability remains the same for the second draw, i.e., 5/6. Finally, the probability of selecting a yellow block on the third draw is 1/6. Therefore, the probability of this sequence is (5/6) * (5/6) * (1/6) = 25/216.
Option 2: Selecting five purple blocks followed by a yellow block (PPPPP-Y)
Similarly, we need to calculate the probability of this sequence. The probability of selecting a purple block on each draw is 5/6, as before. Since we return the block to the bag after each draw, the probability remains the same for all five draws. Finally, the probability of selecting a yellow block on the sixth draw is 1/6. Therefore, the probability of this sequence is (5/6) * (5/6) * (5/6) * (5/6) * (5/6) * (1/6) = 625/7776.
Comparing the probabilities, we see that 25/216 is greater than 625/7776. Therefore, the sequence PPY (two purple blocks followed by a yellow block) is more likely than PPPPP-Y (five purple blocks followed by a yellow block).
In summary, the sequence PPY (two purple blocks followed by a yellow block) is more likely than PPPPP-Y (five purple blocks followed by a yellow block) based on the given probabilities. The key is to calculate the probability of each event and compare them to determine the relative likelihood.