Leap Year Basics
A leap year has 366 days.
To find the number of weeks and remaining days: $366 \div 7 = 52$ with a remainder of 2.
This means a leap year consists of 52 full weeks plus 2 extra days.
Possible Day Combinations
These 2 extra days determine which days of the week occur 53 times. Since the days follow each other sequentially, the two extra days must be consecutive days of the week.
There are 7 possible combinations for these two consecutive days:
- Sunday and Monday
- Monday and Tuesday
- Tuesday and Wednesday
- Wednesday and Thursday
- Thursday and Friday
- Friday and Saturday
- Saturday and Sunday
Each of these 7 combinations is equally likely, so each has a probability of $\frac{1}{7}$.
Favorable Outcomes
We are interested in the probability that the year contains either 53 Sundays or 53 Mondays.
Let's identify the combinations where this occurs:
- 53 Sundays: This happens if one of the extra days is a Sunday. The possible pairs are (Saturday, Sunday) or (Sunday, Monday).
- 53 Mondays: This happens if one of the extra days is a Monday. The possible pairs are (Sunday, Monday) or (Monday, Tuesday).
The event "53 Sundays OR 53 Mondays" occurs if the pair of extra days is:
- (Sunday, Monday): Both Sunday and Monday occur 53 times.
- (Monday, Tuesday): Monday occurs 53 times.
- (Saturday, Sunday): Sunday occurs 53 times.
These are 3 distinct favorable outcomes out of the 7 possible equally likely outcomes.
Calculating the Probability
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
Probability = $\frac{3}{7}$
Important Questions from Probability of Random Experiments
A, B, C and D are mutually exclusive and exhaustive events.
If 2P(A) = 3P(B) = 4P(C) = 5P(D), then what is 77P(A) equal to ?
A fair coin is tossed 6 times. What is the probability of getting a result in the 6t h toss which is different from those obtained in the first five tosses ?
Two cards are drawn successively without replacement from a well-shuffled pack of 52 cards. The probability of drawing two aces is
A biased coin with the probability of getting head equal to \(\frac{1}{4}\) is tossed five times. What is the probability of getting tail in all the first four tosses followed by head ?
Three dice are thrown. What is the probability that each face shows only multiples of 3 ?
