Jenny places 100 pennies on a table, with 30 showing heads and 70 showing tails. She chooses 40 distinct pennies uniformly at random and turns them over. That is, if a chosen penny was showing heads, she turns it to show tails; if a chosen penny was showing tails, she turns it to show heads. After this process, what is the expected number of pennies showing heads?
can give the working process?
To find the expected number of pennies showing heads after Jenny flips 40 distinct pennies, we can use the concept of linearity of expectation.
Let's consider the expectation for a single penny. The probability that a chosen penny starts showing heads is 30/100 since there are initially 30 pennies showing heads out of 100. Similarly, the probability that a chosen penny starts showing tails is 70/100.
When Jenny flips the penny, there are two possibilities:
1. If the chosen penny starts showing heads, then it will end up showing tails after the flip.
2. If the chosen penny starts showing tails, then it will end up showing heads after the flip.
Since Jenny flips each penny independently, the probability of each of these two possibilities is equal. Therefore, the expected number of pennies showing heads after the flip is (30/100) * (0) + (70/100) * (1), which simplifies to 7/10.
Now, since the flips for each penny are independent, we can simply multiply the expected value for a single penny by the number of distinct pennies Jenny flips. In this case, since Jenny flips 40 distinct pennies, the expected number of pennies showing heads is (7/10) * 40 = 28.
Thus, the expected number of pennies showing heads after Jenny flips 40 distinct pennies is 28.