You are attempting to value a put option with an exercise price of $$105$ and one year to expiration. The underlying stock pays no dividends, its current price is $$105,$ and you believe it has a $50\%$ chance of increasing to $$127$ and a $50\%$ chance of decreasing to $$83.$ The risk$-$free rate of interest is $10\%.$ Calculate the value of a put option with exercise price $$105.$ Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

Answer is complete but not entirely correct.

$\backslash$table$[[$Value of a put option,$$0\mathrm{.}47ox$