
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

The fare in a minibus is 75 rubles. Fifteen passengers board one by one and do not exchange money among themselves. There is a probability of 0.5 that a passenger gives the driver a 100-ruble bill and receives 25 rubles in change with coins of 5 or 10 rubles, and a probability of 0.5 that a passenger gives the driver a 50-ruble bill and 25 rubles in coins.

a) Given that the driver had no coins before the trip, what is the probability that the driver will be able to immediately give change to each passenger who pays with a 100-ruble bill?

b) How much money in coins should the driver have before the trip to have at least a 0.95 probability of being able to immediately give change to all passengers who pay with a 100-ruble bill?

Present the answer in LaTex format: \boxed{Your answer}
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