Question

The English alphabet has 26 letters. There are 6 vowels. (a, e, i, o, u, and sometimes y). Suppose we randomly select 8 letters from the alphabet without replacement. Let X = the number of vowels chosen (including y as a vowel). a. How many possible ways are there to select the 8 out of 26 letters (order does not matter) without replacement? 1562275 b. What is the probability that X = 2 0.3721 c. What is the probability that X=1? 0.2977 d. What is the probability that X <3? 0.0806 e. What is the expected value of X? 1.8462 f. What is the standard deviation of X? 0.6448 g. What is the probability that X is within 1.1 standard deviations of its expected value? 0.7194 h Execute the R command "sample(letters, 8)". How many vowels did you get? 1 Add any comments below.

can you try these I do not know if they are correct
(from e-h mainly)
please check the others if you have the time to

Answer 1

Ans: we ways 26c 8 (b) PCT 22 2] = p (sake eting Total letters = 26 Vowels - G can select s letters from 26 letters in 260g 26! 15622-15 Ways 8!(26-8)! 2 vowels out of s letters) GC₂ * 2016 15 * 38760 -0.3721 1562275 26 ce = P( selecting frowel GC,* 2007 6* *7520 03:13 (6 P(x=1)= out of 8 letters s ) 2602 1562275 0.2977 (d). Plac 3)= P(1=0)+P(x+1)+P(=2) Pla =0) = 6o* 2008 0.08OG now) 2608 E(2) = 22 p (X=) 2 3 G 4 X: 0 B P6 P(X=a): 00806 0.2974 0. 3:121 B P4 how P3 = 6C₃42065 = 0.19847920.1985 2608 Gcy 2004 Py 0.0465 2GC8 GG + 2063 Ps 0.00438 -0.00438 2608 GCG & 2002 0.00012 PG 2608 E(X) - Sap(x=2) So, 1.846

Standard deviation of a , Sp(a) =VE()-CECAJE ECA) = 4.430 SD(X) = 1.011 (3). Pleca) T15D(X)) < < €14)+ 1.1(socx))] (1.011) <1.846 +1 (1.011)] = P(1=1) +P ( 22) P[ 0.733 C 2 ep1-846 -1.1 < X < 9.958 )) 2.968) 0.297 Ft 0.37721 0.6698 ch > Sample (letters ; 8) [2] "y u W u u Қ d wouz"