1. Papa Tom Smurf is a smurf-player. He wants to take two smurfettes out on a date. There are 3 blue smurfettes, 5 navy smurfettes and 2 turquoise smurfettes.
a) What’s the probability that the smurfettes he dates are of different colours? (4)
b) If he instead takes three smurfettes out on a date, what’s the probability that they are all of the same colour? (3)
I am extremely confused with probability, can someone please help?
Sure, I can help you understand probability and solve these problems step by step.
Probability is a mathematical concept used to measure the likelihood of an event happening. It is denoted by a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
Let's start by solving problem a) - the probability that the smurfettes Papa Tom dates are of different colors.
To solve this, we need to know the total number of possible outcomes (cases) and the number of favorable outcomes (desired cases). In this case, the total number of possible outcomes is the total number of ways Papa Tom can choose two smurfettes out of all the available smurfettes.
The total number of possible outcomes can be calculated using the combination formula: nCr = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items you want to choose. In our case, n is the total number of smurfettes (which is 3 + 5 + 2 = 10), and r is 2.
So the total number of possible outcomes, denoted as C(n, r) or "10 choose 2," is calculated as follows:
C(10, 2) = 10! / (2! * (10-2)!) = 10! / (2! * 8!) = (10 * 9) / (2 * 1) = 45.
Now, let's calculate the number of favorable outcomes (the desired cases). In this case, favorable outcomes are when Papa Tom chooses two smurfettes of different colors.
There are three different color options to choose from for the first smurfette, and since we want different colors for the second smurfette, there will be eight options left. Therefore, the number of favorable outcomes is 3 * 8 = 24.
Now we can find the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Favorable Outcomes / Total Outcomes
Probability = 24 / 45 ≈ 0.5333 (rounded to four decimal places).
So the probability that the smurfettes Papa Tom dates are of different colors is approximately 0.5333 or 53.33% (rounded to two decimal places).
Now, let's move on to problem b) - the probability that three smurfettes Papa Tom dates are all of the same color.
Again, we need to calculate the total number of possible outcomes and the number of favorable outcomes. The calculations will be similar to problem a).
The total number of possible outcomes is now the total number of ways Papa Tom can choose three smurfettes out of all the available smurfettes, which is C(10, 3) = 10! / (3! * (10-3)!) = 10! / (3! * 7!) = 120.
The number of favorable outcomes is when Papa Tom chooses three smurfettes of the same color. Since there are three different colors available, the number of favorable outcomes is 3 (one for each color).
Therefore, the probability of all three smurfettes being of the same color is 3 / 120 = 0.025 or 2.5% (rounded to one decimal place).
I hope this explanation helps! Let me know if you have any further questions.
Blue -- B --- 3
Navy -- N -- 5
Turquoise -- T -- 2
a) his choices could be BN, BT, OR NT
prob(Blue Navy) = C(3,1) C(5,1)/C(10,2) = 3*5/45 = 1/3
prob(B T) = C(3,1) C(2,1)/C(10,2) = 6/45 = 2/15
prob(N T) = ...
add them up
or
Prob(B N) = (3/10) (5/9) (2) , times 2 because it could be BN or NB
= 30/90 = 1/3
etc for the other two
b) 3 of the same colour, implies it can only be BBB or NNN
combination way:
prob(BBB) = C(3,3)/C(10,3) = 1/120
other way:
prob(BBB) = (3/10)(2/9)(1/8) = 6/720 = 1/120
find NNN in the same way, then add them up