Mixed Strategy Bayesian Nash Equilibrium
The trick for finding a mixed strategy Nash Equilibrium is that given everyone else's strategies, all players will be indifferent between each of the options their randomizing over (ie. those options will yield the same payoff). So all you need to do is write an expression relating each player's expected payoffs for each strategy, and solve for the frequencies. Letting x represent the probability that player j does the project, low-cost player i's payoff for doing the project is 5, while their expected payoff for not doing the project is 10x. So you know in the Nash Equilibrium, 5=10x. The incomplete information makes the problem slightly more complicated: player i may expect j to complete the project half the time, the task is now finding j's probability of completing the project given that they are low-cost (not too difficult, since high cost players will never do the project). I'll leave the rest to you.