Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 2 times and lost 8 times.

1. Car
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Goat
8. Car
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 317 times and lost 683 times.

1. Car
2. Goat
3. Car
4. Goat
5. Car
6. Goat
7. Goat
8. Car
9. Goat
10. Goat
11. Car
12. Goat
13. Goat
14. Goat
15. Goat
16. Goat
17. Goat
18. Goat
19. Car
20. Goat
21. Goat
22. Goat
23. Goat
24. Goat
25. Goat
26. Goat
27. Car
28. Goat
29. Car
30. Goat
31. Goat
32. Goat
33. Car
34. Goat
35. Goat
36. Car
37. Goat
38. Goat
39. Goat
40. Car
41. Car
42. Goat
43. Goat
44. Goat
45. Goat
46. Goat
47. Goat
48. Goat
49. Goat
50. Goat
51. Car
52. Goat
53. Goat
54. Goat
55. Car
56. Goat
57. Goat
58. Goat
59. Goat
60. Car
61. Goat
62. Goat
63. Goat
64. Goat
65. Car
66. Goat
67. Goat
68. Goat
69. Goat
70. Goat
71. Goat
72. Goat
73. Car
74. Goat
75. Car
76. Car
77. Car
78. Goat
79. Goat
80. Car
81. Car
82. Goat
83. Goat
84. Goat
85. Goat
86. Goat
87. Car
88. Goat
89. Goat
90. Goat
91. Goat
92. Goat
93. Car
94. Goat
95. Goat
96. Goat
97. Goat
98. Goat
99. Car
100. Car
101. Goat
102. Goat
103. Goat
104. Goat
105. Car
106. Goat
107. Goat
108. Goat
109. Car
110. Car
111. Goat
112. Goat
113. Car
114. Goat
115. Car
116. Goat
117. Goat
118. Car
119. Goat
120. Goat
121. Goat
122. Car
123. Goat
124. Car
125. Goat
126. Goat
127. Car
128. Car
129. Goat
130. Goat
131. Goat
132. Goat
133. Car
134. Goat
135. Goat
136. Goat
137. Goat
138. Goat
139. Goat
140. Car
141. Goat
142. Car
143. Car
144. Goat
145. Goat
146. Goat
147. Goat
148. Goat
149. Goat
150. Car
151. Goat
152. Goat
153. Goat
154. Goat
155. Goat
156. Goat
157. Goat
158. Car
159. Car
160. Car
161. Goat
162. Car
163. Goat
164. Goat
165. Car
166. Car
167. Car
168. Goat
169. Goat
170. Goat
171. Car
172. Car
173. Goat
174. Goat
175. Goat
176. Goat
177. Goat
178. Goat
179. Goat
180. Car
181. Goat
182. Car
183. Goat
184. Goat
185. Goat
186. Car
187. Car
188. Goat
189. Car
190. Car
191. Goat
192. Goat
193. Car
194. Goat
195. Goat
196. Goat
197. Goat
198. Car
199. Goat
200. Goat
201. Goat
202. Car
203. Goat
204. Car
205. Goat
206. Car
207. Car
208. Goat
209. Car
210. Car
211. Goat
212. Goat
213. Goat
214. Goat
215. Goat
216. Car
217. Goat
218. Goat
219. Goat
220. Car
221. Goat
222. Goat
223. Goat
224. Goat
225. Goat
226. Goat
227. Goat
228. Goat
229. Car
230. Goat
231. Goat
232. Car
233. Goat
234. Goat
235. Goat
236. Goat
237. Goat
238. Car
239. Goat
240. Goat
241. Car
242. Goat
243. Goat
244. Goat
245. Goat
246. Goat
247. Goat
248. Goat
249. Goat
250. Car
251. Car
252. Goat
253. Car
254. Car
255. Goat
256. Goat
257. Goat
258. Goat
259. Car
260. Car
261. Car
262. Car
263. Goat
264. Car
265. Goat
266. Car
267. Goat
268. Car
269. Goat
270. Car
271. Goat
272. Goat
273. Goat
274. Car
275. Car
276. Goat
277. Goat
278. Goat
279. Car
280. Goat
281. Goat
282. Goat
283. Goat
284. Goat
285. Car
286. Goat
287. Car
288. Car
289. Goat
290. Goat
291. Goat
292. Goat
293. Goat
294. Goat
295. Car
296. Car
297. Car
298. Goat
299. Goat
300. Goat
301. Goat
302. Car
303. Goat
304. Goat
305. Goat
306. Goat
307. Car
308. Car
309. Car
310. Goat
311. Car
312. Car
313. Goat
314. Goat
315. Goat
316. Car
317. Goat
318. Car
319. Goat
320. Goat
321. Car
322. Goat
323. Goat
324. Car
325. Car
326. Goat
327. Goat
328. Goat
329. Goat
330. Goat
331. Goat
332. Goat
333. Car
334. Car
335. Goat
336. Goat
337. Goat
338. Goat
339. Goat
340. Goat
341. Car
342. Goat
343. Goat
344. Car
345. Goat
346. Goat
347. Goat
348. Car
349. Goat
350. Car
351. Car
352. Car
353. Goat
354. Goat
355. Goat
356. Car
357. Goat
358. Goat
359. Goat
360. Goat
361. Goat
362. Car
363. Goat
364. Goat
365. Goat
366. Car
367. Goat
368. Goat
369. Goat
370. Goat
371. Goat
372. Goat
373. Goat
374. Goat
375. Car
376. Car
377. Goat
378. Goat
379. Goat
380. Goat
381. Car
382. Car
383. Goat
384. Goat
385. Goat
386. Goat
387. Goat
388. Car
389. Goat
390. Car
391. Car
392. Car
393. Car
394. Car
395. Goat
396. Car
397. Car
398. Goat
399. Goat
400. Goat
401. Goat
402. Goat
403. Goat
404. Car
405. Goat
406. Car
407. Goat
408. Goat
409. Car
410. Car
411. Car
412. Car
413. Car
414. Car
415. Goat
416. Goat
417. Goat
418. Car
419. Goat
420. Car
421. Goat
422. Goat
423. Goat
424. Goat
425. Goat
426. Goat
427. Goat
428. Goat
429. Goat
430. Goat
431. Goat
432. Goat
433. Goat
434. Goat
435. Car
436. Car
437. Goat
438. Car
439. Goat
440. Car
441. Car
442. Goat
443. Goat
444. Goat
445. Car
446. Goat
447. Car
448. Goat
449. Goat
450. Car
451. Goat
452. Car
453. Goat
454. Goat
455. Goat
456. Car
457. Car
458. Car
459. Car
460. Car
461. Goat
462. Car
463. Goat
464. Goat
465. Goat
466. Car
467. Goat
468. Goat
469. Goat
470. Goat
471. Car
472. Goat
473. Car
474. Goat
475. Car
476. Goat
477. Goat
478. Goat
479. Goat
480. Goat
481. Goat
482. Car
483. Goat
484. Goat
485. Goat
486. Car
487. Goat
488. Goat
489. Goat
490. Car
491. Car
492. Car
493. Car
494. Car
495. Goat
496. Goat
497. Goat
498. Goat
499. Goat
500. Goat
501. Goat
502. Goat
503. Goat
504. Goat
505. Car
506. Goat
507. Goat
508. Goat
509. Goat
510. Car
511. Car
512. Goat
513. Car
514. Car
515. Goat
516. Goat
517. Goat
518. Goat
519. Goat
520. Goat
521. Car
522. Goat
523. Car
524. Goat
525. Goat
526. Goat
527. Goat
528. Car
529. Goat
530. Goat
531. Goat
532. Goat
533. Goat
534. Goat
535. Goat
536. Goat
537. Car
538. Car
539. Goat
540. Car
541. Car
542. Goat
543. Goat
544. Goat
545. Goat
546. Goat
547. Goat
548. Goat
549. Goat
550. Goat
551. Car
552. Car
553. Goat
554. Goat
555. Goat
556. Goat
557. Goat
558. Goat
559. Goat
560. Goat
561. Car
562. Car
563. Car
564. Goat
565. Goat
566. Car
567. Car
568. Goat
569. Goat
570. Goat
571. Goat
572. Goat
573. Car
574. Car
575. Goat
576. Goat
577. Car
578. Goat
579. Goat
580. Goat
581. Car
582. Goat
583. Car
584. Goat
585. Goat
586. Goat
587. Goat
588. Car
589. Car
590. Goat
591. Goat
592. Car
593. Goat
594. Goat
595. Car
596. Goat
597. Goat
598. Car
599. Goat
600. Goat
601. Goat
602. Goat
603. Goat
604. Goat
605. Goat
606. Car
607. Car
608. Goat
609. Goat
610. Goat
611. Car
612. Goat
613. Goat
614. Goat
615. Car
616. Goat
617. Goat
618. Goat
619. Goat
620. Goat
621. Goat
622. Goat
623. Goat
624. Goat
625. Goat
626. Goat
627. Goat
628. Goat
629. Goat
630. Goat
631. Goat
632. Goat
633. Goat
634. Goat
635. Car
636. Car
637. Goat
638. Goat
639. Goat
640. Car
641. Car
642. Car
643. Goat
644. Goat
645. Goat
646. Goat
647. Car
648. Car
649. Goat
650. Goat
651. Car
652. Goat
653. Goat
654. Goat
655. Goat
656. Car
657. Car
658. Car
659. Goat
660. Car
661. Goat
662. Goat
663. Goat
664. Car
665. Car
666. Goat
667. Goat
668. Goat
669. Goat
670. Goat
671. Car
672. Goat
673. Goat
674. Car
675. Goat
676. Goat
677. Car
678. Car
679. Goat
680. Goat
681. Car
682. Goat
683. Goat
684. Car
685. Car
686. Car
687. Goat
688. Goat
689. Goat
690. Goat
691. Car
692. Goat
693. Goat
694. Car
695. Goat
696. Goat
697. Goat
698. Goat
699. Car
700. Goat
701. Car
702. Car
703. Goat
704. Car
705. Goat
706. Goat
707. Goat
708. Goat
709. Goat
710. Car
711. Car
712. Goat
713. Goat
714. Car
715. Car
716. Goat
717. Car
718. Goat
719. Goat
720. Car
721. Goat
722. Car
723. Goat
724. Goat
725. Car
726. Car
727. Car
728. Goat
729. Car
730. Car
731. Car
732. Goat
733. Goat
734. Goat
735. Goat
736. Goat
737. Goat
738. Goat
739. Goat
740. Car
741. Goat
742. Car
743. Car
744. Goat
745. Car
746. Goat
747. Goat
748. Goat
749. Car
750. Goat
751. Goat
752. Car
753. Car
754. Goat
755. Goat
756. Goat
757. Goat
758. Car
759. Car
760. Goat
761. Car
762. Goat
763. Goat
764. Goat
765. Goat
766. Goat
767. Goat
768. Goat
769. Goat
770. Goat
771. Goat
772. Goat
773. Goat
774. Goat
775. Car
776. Goat
777. Goat
778. Goat
779. Goat
780. Car
781. Car
782. Goat
783. Goat
784. Goat
785. Car
786. Goat
787. Goat
788. Goat
789. Goat
790. Goat
791. Car
792. Goat
793. Goat
794. Car
795. Goat
796. Car
797. Goat
798. Car
799. Car
800. Goat
801. Goat
802. Car
803. Car
804. Goat
805. Goat
806. Goat
807. Goat
808. Car
809. Goat
810. Car
811. Car
812. Car
813. Goat
814. Car
815. Goat
816. Goat
817. Goat
818. Goat
819. Goat
820. Car
821. Car
822. Goat
823. Goat
824. Goat
825. Goat
826. Goat
827. Goat
828. Car
829. Goat
830. Car
831. Goat
832. Goat
833. Car
834. Car
835. Goat
836. Goat
837. Car
838. Goat
839. Goat
840. Car
841. Goat
842. Goat
843. Goat
844. Car
845. Goat
846. Car
847. Goat
848. Goat
849. Goat
850. Goat
851. Goat
852. Goat
853. Goat
854. Goat
855. Car
856. Goat
857. Goat
858. Goat
859. Goat
860. Goat
861. Goat
862. Car
863. Car
864. Goat
865. Goat
866. Car
867. Car
868. Car
869. Goat
870. Goat
871. Goat
872. Goat
873. Car
874. Goat
875. Goat
876. Goat
877. Goat
878. Goat
879. Goat
880. Goat
881. Goat
882. Car
883. Car
884. Car
885. Car
886. Car
887. Goat
888. Car
889. Goat
890. Goat
891. Car
892. Car
893. Goat
894. Goat
895. Car
896. Goat
897. Goat
898. Goat
899. Goat
900. Goat
901. Goat
902. Goat
903. Car
904. Goat
905. Goat
906. Car
907. Goat
908. Goat
909. Car
910. Goat
911. Goat
912. Goat
913. Car
914. Goat
915. Goat
916. Goat
917. Car
918. Goat
919. Goat
920. Goat
921. Car
922. Goat
923. Car
924. Car
925. Car
926. Goat
927. Goat
928. Car
929. Goat
930. Goat
931. Car
932. Goat
933. Car
934. Goat
935. Goat
936. Goat
937. Goat
938. Car
939. Goat
940. Goat
941. Goat
942. Goat
943. Car
944. Goat
945. Goat
946. Car
947. Car
948. Goat
949. Goat
950. Goat
951. Goat
952. Car
953. Goat
954. Goat
955. Car
956. Goat
957. Goat
958. Goat
959. Goat
960. Goat
961. Goat
962. Goat
963. Goat
964. Goat
965. Car
966. Goat
967. Car
968. Goat
969. Car
970. Goat
971. Car
972. Goat
973. Car
974. Goat
975. Goat
976. Goat
977. Goat
978. Goat
979. Goat
980. Goat
981. Goat
982. Goat
983. Goat
984. Goat
985. Goat
986. Car
987. Car
988. Car
989. Car
990. Car
991. Car
992. Goat
993. Goat
994. Goat
995. Goat
996. Goat
997. Goat
998. Goat
999. Car
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 117 times and lost 883 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 34 times and lost 66 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}