Probability Events - Total Probability of An Event Happening

Question

I can't seem to work out the below probability question. Wonder if anyone can help? Thanks in advance!

"10 people have made a booking for a shuttle transfer and the departure time is 10am. If all customers arrive early, the shuttle will depart earlier. The shuttle will wait until 10:10am if any customers are late. If the probability of customers arriving 10 min earlier is 0.1, 5 min earlier is 0.2, right at 10am is 0.5, late for 5 min is 0.1, late for 10 min is 0.05, and late for over 10min is 0.05. What is the probability for the shuttle to depart on or before 10am?"

Answer 1

This can be expressed in a binomial distribution if you let p=0.8 be the probability that an individual arrives on time (0.1+0.2+0.5), and let n=10 denote the number of trials.

The probability mass function (PMF) of a binomial distribution is defined as

That is, the probability of observing k successes in n trials with probability p.

Now, the probability whether the shuttle will depart on (or before) time, is equivalent to the probability that all individuals arrive at least on time, that is P(X = k) where k=10 successes.

Using the above PMF we obtain