Binomial Theorem

Rarely Tested

From the binomial theorem, we know that

$$\left(x+y\right)^n=^nC_0x^ny^0+^nC_1x^{n-1}y^1+^nC_2x^{n-2}y^2+.\ .\ .\ .\ .+^nC_nx^0y^n$$

with $$T_r=^nC_rx^{n-r}y^r$$

Question 1

Roger Federer and Novak Djokovic are playing the tie-breaker of 5th set in the Australian open final. The player who reaches 7 points first in the tie-breaker wins the set. The prior probability that Roger will win a point against Djokovic is 0.7. What is the probability that Federer wins the tie-breaker exactly on the 10th point of the tie-breaker?

Question 2

Raunak is practicing shooting on a target. The probability he will correctly hit a target in an individual shot is 1/5. In a round of 10 shots, what is the probability that the target is hit at least twice?

Question 3

If the coefficient of $$x^8$$ in the expansion $$(px^2+\frac{1}{qx})^{13}$$ equals the coefficient of $$x^{-8}$$ in $$(px-\frac{1}{qx^2})^{13}$$, what is the relationship between p and q?