Consider a $2-$period option with an exercise price of $$50.$ The current stock price is $$48,$ and the risk free rate is $2\%.$ If the stock moves up$,$it increases in value by $20\%.$ If the stock moves down, It decreases in value by $10\%$

$1)$ What is the probability of an upward movement?

$2)$ What is the value of a call option?