A veteran public official is up for reelection and is concerned about ~p , the proportion of the votes cast that will be for him. In the past, he has run for office 20 times; he received 48 percent of the vote four times, 50 percent of the vote four times, 52 percent of the vote five times, 54 percent of the vote four times, 56 percent of the vote twice, and 58 percent of the vote once. In the upcoming election, however, his opponent is young, vigorous, and has a large campaign budget. As a result, the veteran official feels that the past frequencies slightly overstate his chances. He feels that this opponent will obtain about 2 percent more of the votes than his past opponents, so that his prior distribution consists of the probabilities P( ~p = 0.46) = 4/20, P( ~p = 0.48) = 4/20, P( ~p = 0.50) = 5/20, and so on. He then conducts a small survey of 100 voters, 49 of whom state that they will vote for him and 51 of whom state that they will vote for his young opponent. Find the posterior distribution of ~p and use this distribution to calculate the expected proportion of votes the veteran public official will obtain.