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 4 times and lost 6 times.

1. Goat
2. Goat
3. Car
4. Goat
5. Goat
6. Goat
7. Car
8. Car
9. Car
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 312 times and lost 688 times.

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

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 96 times and lost 904 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 27 times and lost 73 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;
	}
}