Will baseball’s winning streak record be broken?

I came across this story in ESPN: Which of baseball’s most unbreakable records might actually get broken in 2019?
You can pass the story, unless you are really baseball statistics nerds, or don’t have something better to do, like me. But there is an interesting probability question there.
The longest winning streak in MLB history was 26 wins in a row, and this record was set by the Giants in 1916. What is the probability that this record will be broken?
The author, Sam Miller, argues that the chance is about 1 in 250. His reasoning goes along these lines: First he assumes that the best team will win about 100-110 games out of the season’s 162 games. The next assumption is that the probability of winning is the same for each game, therefore this probability ranges in 0.62 to 0.68. He does not state the third assumption explicitly, but you can’t do the math without it: games are independent.
All these assumptions translate to a series of 162 Bernoulli trial, with success probability of about 0.65 (plus/minus 0.03). So, what is the probability of getting a streak of at least 27 successes? Can you do the calculations?

This post was originally posted at Statistically Speaking.