Shortcut -Probability

Question:

What is the probability of selecting 2 spades from a pack of 52 cards?

Total number of ways of selecting 2 cards from 52 = 52C2 = 1275
Total ways of selecting 2 spades from 13 spades = 13C2 = 78
Probability = 78/1275

Question:

What is the probability of selecting 2 students from a class of 10 boys and 8 girls, where both the selected students are boys?

Total number of ways of selecting 2 students from 18
= 18C2 = 153
Total ways of selecting 2 boys from 10 boys
= 10C2 = 45
Probability = 45/153

Question:

What is the probability of getting 4 heads when 7 coins are tossed?

Total number of results = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 28 = 128
When the required results need 4 heads, the remaining three will be tails.
The results will be the different arrangement of the following:
“HHHHTTT”
The number of ways in which the above can be arranged is
= 7!/(4! x 3!)
= 5040/144
= 35
Probability = 35/128

Question:

What is the probability of getting a sum which is equal to prime number from the results obtained by throwing two dice with six faces numbered 1 to 6 on the faces?

The above table shows the different sums obtained from different results.
Total number of results = 36.
Expecting result = 15. (Highlighted with the shade).
Probability = 15/36

Question:

What is the probability of selecting 4 balls from a bag that has 5 red balls and 6 black balls where the selection has at least 1 red ball?

The different cases are:
1 red and 3 black = 5C1 x 6C3 = 100
2 red and 2 black = 5C2 x 6C2 = 150
3 red and 1 black = 5C3 x 6C1 = 60
4 red = 5C4 =5
Number of ways of selecting 4 balls with at least 1 red ball:
= 100 + 150 + 60 + 5 = 315
Total ways of selecting 4 balls without any condition:
= 11C4 = 330
Probability = 315/330
Alternate method:
Probability = 1 – probability of selecting no red balls
= 1 – probability of selecting 4 black balls
Probability = 1 – (6C4/11C4)
= 1 – (15/330)
= 315/330