The problem is believing the fairness of the toss.

The odds with a fair toss are obviously stacked to prefer taking the gamble, hence inducing all credulous rationalists to take the bet.

It is rare for anyone to give away £50k, and even rarer to give away £1m. There is no realistic scenario where such a bet would be offered ... Almost all of us have been to the fairground (yes, the analysis is very different when you have to pay to play).

Most people instinctively reason that the actual probability of a win is close to 0: exactly zero in a one shot game, but perhaps some epsilon in a repeated trial in the fairground, so as to advertise a verified previous winner.

[I am willing to bet that fairground stalls give a giant cuddly toy to an employee to walk around the ground, or perhaps throw a game to an employee in front of a crowd of onlookers, to suggest that there is an epsilon of hope, even when there is none in reality].

Everyone's prior instincts should be conditioned to take the smaller 'sure' reward, since that requires strictly fewer trustworthy steps to success (one payout, compared to both a non-zero toss and a payout).

The irony is that in a thought experiment, no game is actually played, so the expected return is always what the wise fair-goer assumed - precisely zero.

In a UK survey of ~4500 adults, 73% said they would take a certain £50k over a 50/50 chance of £1m. For women, it was 82%.

via https://www.bbc.co.uk/news/articles/cn0njd4e2evo

If your get paid more than 50k net, then 50k isn't that much and 50/50 chance of 1M is a much more attractive proposition. If you get paid significantly less then a guaranteed 50k is a good deal.

What would you do?

The expected value of the 50/50 is $500,000.

It's mathematically the best choice.