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?
YPPYPP
YYYPPP
YYYPP
YPPYPP
In order to determine which sequence of selected blocks is more likely, we need to analyze the probabilities associated with each sequence. Let's break it down:
Option 1: YPPYPP
The probability of selecting a yellow block (Y) on any single draw is 1 out of 6. The probability of selecting a purple block (P) is 5 out of 6. In this sequence, the first draw is a yellow block, which has a probability of 1/6. The second draw is a purple block, and again the probability is 5/6. The third draw is another purple block, with the same probability of 5/6. The fourth draw is a yellow block, which has a probability of 1/6. The fifth draw is a purple block, with a probability of 5/6, and the last draw is another purple block, again with a probability of 5/6.
To calculate the probability of this specific sequence, we multiply the individual probabilities together:
(1/6) * (5/6) * (5/6) * (1/6) * (5/6) * (5/6) = 625/46656
Option 2: YYYPPP
Similarly, let's analyze the probabilities for this sequence. The first three draws are all yellow blocks, each with a probability of 1/6. The next three draws are purple blocks, each with a probability of 5/6. So, to calculate the probability of this sequence:
(1/6) * (1/6) * (1/6) * (5/6) * (5/6) * (5/6) = 125/46656
Option 3: YYYPP
In this sequence, the first three draws are all yellow blocks, each with a probability of 1/6. The next two draws are purple blocks, each with a probability of 5/6. The probability calculation for this sequence is:
(1/6) * (1/6) * (1/6) * (5/6) * (5/6) = 25/7776
Comparing the probabilities:
- Option 1 (YPPYPP): 625/46656
- Option 2 (YYYPPP): 125/46656
- Option 3 (YYYPP): 25/7776
Therefore, the sequence "YPPYPP" (Option 1) has the highest probability out of the given options.