
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 for the minibus is 75 rubles. Fifteen passengers board individually without exchanging money amongst themselves. With a probability of 0.5, a passenger gives the driver a 100-ruble bill and receives 25 rubles in change with coins of 5 or 10 rubles, and with a probability of 0.5, a passenger gives the driver a 50-ruble bill and 25 rubles in coins.

a) Before the trip, the driver had no coins at all. 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 in coins should the driver have before the trip to be able to give change immediately to all passengers who pay with a 100-ruble bill, with a probability of at least 0.95?

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