Doughnuts

On election day in 1992, a California TV station (Channel 7 in the San Diego area) conducted a statewide survey of 2500 voters leaving the polls, and found, among other things, that 51%claimed to have voted for Barbara Boxer and 49%claimed to have voted for her opponent. On the basis of this sample, they declared Boxer the winner.

Problems

1. If you flip a fair coin 2500 times, what (roughly) is the probability that it will come up heads at least 51%of the time?

2. Suppose you have a coin that is biased so that it comes up heads 51%of the time over the long, long haul. You flip this coin 2500 times. What (roughly) is the probability that it comes up heads at least half the time?

3. If Channel 7 had been willing to bet that Boxer would eventually win, would you have taken the bet if they offered odds of 2 to 1? 10 to 1? 100 to 1? dollars to doughnuts?