For many folk, the results of probability calculations are nonintuitive. (Or maybe I should just speak for myself. I'm often surprised.)
The probability of randomly getting 60% right in 140 trials is 0.016. While one might not offer a million dollar bet against those odds, by most standards that is "statistically far from the expected 50%."
First off, your math is wrong. You gave the probability of getting 60 correct, not 60%. The probablity of getting 60%, which is 84 correct, is 0.004.
I don't think either number is statistically far from the expected 50% at all. Look at the chart below. The larger black box is where the probability of getting 60% lies. It's clearly within the meat of curve. If you want to talk about what's "far off" the scale, you need to start looking where you can't even see the line anymore.
You can think of it another way. Suppose you're giving a lecture to 100 freshman about stats. You have them flip a coin 140 times each. You're not going to be very surprised if somebody has 60% of them turn up heads. You might raise an eyebrow, but it's not terribly surprising. If you test this person again, odds are they will get a much lower score. If they get a similar high score, that's when you get excited.
Even then, what does that really mean? Suppose the person says they have a skill that allows them to flip the coin and have it come up heads. Does this mean they can do it 60% of the time? Absolutely not. Remember, we
expect to see heads about 50% of the time anyway. Their "influence" on the coin flip is very small. If the ability were real, it would mean the skill only worked about 5% to 10% of the time (a guess...didn't crunch the numbers).
Stated another way, if the person could influence the flip 60% of the time, then 84 of the flips would be due to their ability, not chance. That leaves the other 56 flips to chance. This means another 28 or so flips will come up heads by chance. That means we would expect 112 heads for a total of 80% for someone who could influence the flip 60% of the time.
If that's hard to grasp intuitively, think about a multiple choice (4 options) test on identifying 100 people by name. Suppose you don't know anybody pictured on the test. You'll get about 25% correct. Does that mean you know 25% of the people? No.
Now let's suppose you know 10% of the people pictured. What will your score be? Will it be 10%? No. You'll get 10 correct due to knowledge. You'll also get 25% of the rest due to guessing. 25% of 90 is 23. Thus, your score will be around 33. So someone who knows 10% of the questions on that test should get about 33% correct.
This, of course, brings us back to what is meant by far off stastically. These small effect sizes get lumped together with similar scores in the meat of the curve. You need a lot of trials before you can be confident that you're seeing a small effect rather than chance.